Area questions in CUET Maths Section B1 take the area under a curve from Section A to circles, parabolas and ellipses, and the parts of them cut off by a line. Each one is a definite integral, set up from a sketch. Learn each rule here, from regions bounded by lines and integrating along y to the areas of circles, parabolas and ellipses and the parts cut off by a line, with worked examples, time-saving shortcuts and the traps CUET sets, then practise on free questions in CUET's own formats, each fully solved.
An oval garden bed is 14 m long and 10 m wide. How much turf covers it? Its edge is an ellipse, and integration finds the area of any region bounded by curves we can write down: slice it into thin strips, find each strip's length from the curves, and add the strips with an integral.
Sketch the region. Use vertical strips when the region sits between a curve and the -axis, and integrate . Use horizontal strips when it sits against the -axis, and integrate . An area is never negative: take the absolute value of each part.
A region bounded by straight lines is often a triangle or trapezium, and the integral must agree with the geometry, which is a good check.
Example 1 · A region against the y-axis
Find the area between , the -axis and the lines and .
Use horizontal strips: .
.
12 square units
In the first quadrant, is , and
The quadrant has area , so the circle has . For a part cut off by a line , integrate from to and double for the half below the axis.
Example 2 · A whole circle
Find, by integration, the area enclosed by .
.
.
square units
Solve for the part you integrate: gives above the axis, and gives beside it. The region of cut off by its latus rectum has area .
Example 3 · A parabola and its latus rectum
Find the area bounded by and the line .
Above the axis ; by symmetry, area .
.
24 square units, which is
Example 4 · A parabola below a line
Find the area bounded by and .
They meet at , and the line lies above: .
.
36 square units
For , the first-quadrant arc is , so the area is .
Example 5 · The garden bed
The bed is the ellipse , in metres. What is its area?
, .
Area .
, about 110 m²
Watch out
The semi-axes are the square roots of the denominators. For they are 7 and 5, not 49 and 25.
Setting up
Circles and ellipses
Parabolas
Next in Section B1, two more ways to solve a differential equation: homogeneous and linear differential equations.
115 questions, grouped by type, easy to hard. Pick an answer and check it. The worked solution opens when you ask for it.
What is the area of the region bounded by the line , the -axis and the lines and ?
AnswerC · 12 square units
.
square units, a trapezium with parallel sides 1 and 7.
Worth noting
9 integrates alone and forgets the strip, .
What is the area of the region between the lines and from to ?
AnswerB · 4.5 square units
On , is the upper line: .
square units.
Worth noting
9 is the area under alone, without taking away the region under .
Which integral gives the area of the region bounded by and the two coordinate axes?
AnswerD ·
The line meets the axes at and , and lies above the -axis between them.
Area .
Worth noting
subtracts the wrong way round and comes out negative.
Using integration, what is the area of the region bounded by the line and the coordinate axes?
AnswerC · 16 square units
The line meets the axes at and ; above the -axis it is .
square units.
Worth noting
32 is the rectangle , twice the triangle; 12 stops the integral at , the -intercept.
What is the area of the region bounded by , the -axis and the lines and ?
AnswerA · 6 square units
.
square units.
Worth noting
8 is the rectangle of width 2 and height 4, the line's height at used across the whole strip.
The region bounded by the line , the -axis and the line is found with vertical strips. Between which limits is integrated?
AnswerD · to
The line meets the -axis where , that is at .
The region runs from there to the line : square units.
Worth noting
Starting at treats the -axis as a side, but this region ends where the line meets the -axis.
For , what is the area of the triangle bounded by the line and the coordinate axes?
AnswerD ·
The line meets the axes at and , and lies above the -axis between them.
square units.
Worth noting
is the square on the two legs; the triangle is half of it.
A region lies in the first quadrant. It is bounded by the -axis, the lines and , and the line .
At which points does the line meet the lines and ?
AnswerA · and
On : . On : .
So the slanting side runs from to .
Worth noting
and swap the coordinates: a point on has -coordinate 1.
What is the area of the region bounded by the line , the -axis and the lines and ?
AnswerB · 4 square units
The region lies between the curve and the -axis, so integrate along : .
square units.
Worth noting
8 forgets the factor in .
Read the two statements and choose the correct option.
I. The area between a curve , the -axis and , is .
II. An area found by integration can be negative.
AnswerA · Only I
I is the horizontal-strip version of the area formula. True.
II is false: an integral can be negative, but an area is its absolute value.
Using integration, what is the area of the triangle bounded by , and the -axis?
AnswerB · 16 square units
The lines meet at and cut the -axis at 0 and 8.
square units.
Worth noting
32 uses base height without the .
What is the area of the region bounded by , the -axis and the -axis?
AnswerC · 4.5 square units
The region lies below the -axis, for : .
An area is never negative: square units.
Worth noting
is the integral; the area is its absolute value.
What is the area of the region between , the -axis and the lines and ?
AnswerC · 6 square units
Horizontal strips: .
square units.
Worth noting
4 integrates alone, leaving out the strip of width 1.
What is the area of the region bounded by and ?
AnswerB · 4 square units
They meet at ; the line lies above: .
square units.
Worth noting
8 is the rectangle ; the region is the triangle above the V, half of it.
Using integration, what is the area of the triangle with vertices , and ?
AnswerB · 8 square units
The sides from the top vertex are for and for .
square units.
Worth noting
4 integrates only one of the two sides, stopping at .
What is the area of the region bounded by , the -axis and the lines and ?
AnswerC · 5 square units
for and for , so split at 0: .
square units.
Worth noting
4 is , which reads as : the small triangle left of the -axis is taken away instead of added.
For , the region bounded by , the -axis and the line has the same area as the triangle bounded by and the coordinate axes. What is ?
AnswerB ·
, and the triangle has area .
gives .
Worth noting
2 takes as , dropping the from .
Assertion (A): The area enclosed by is 18 square units.
Reason (R): The region is symmetric about both axes, and its part in the first quadrant is the triangle bounded by and the axes.
AnswerA · Both A and R are true, and R is the correct explanation of A.
In the first quadrant is , and . R is true.
By that symmetry the area is : A is true, and R is the reason.
Worth noting
The first-quadrant triangle alone is ; the symmetry in R is what multiplies it by 4.
Read the two statements and choose the correct option.
I. The region bounded by , the -axis and the line has the same area as the region bounded by , the -axis and the line .
II. is the area of the region bounded by , the -axis and the lines and .
AnswerA · Only I
I: and ; the two triangles together fill the rectangle . True.
II: , but the region is two triangles of area each, 1 in all. False.
Worth noting
Taking II as true lets the triangle below the axis cancel the one above; an area adds their sizes.
For the region bounded by the line , the -axis and the line , which of the following statements are true?
I. It is the triangle with vertices , and .
II. Its area is .
III. With horizontal strips, its area is .
IV. With horizontal strips, its area is .
AnswerD · I, II and III only
The line meets the -axis at and at : a triangle of area , and . I and II hold.
A strip at height runs from to , length : , so III holds. IV measures the strip from the -axis to the line and gives : false.
Worth noting
Leaving out III misses that horizontal strips, each from the line to , give the same 8.
Using horizontal strips, which integral gives the area of the triangle bounded by the lines , and the -axis?
AnswerD ·
The lines meet where , at . A strip at height runs from to .
Its length is , so the area is square units.
Worth noting
measures each strip from the -axis instead of from the line .
A region lies in the first quadrant. It is bounded by the -axis, the lines and , and the line .
Which integral gives the area of ?
AnswerD ·
lies against the -axis, so use horizontal strips: at height a strip runs from to .
The strips run from to : .
Worth noting
starts at the -axis, but the lower side of is the line .
A region lies in the first quadrant. It is bounded by the -axis, the lines and , and the line .
What is the area of ?
AnswerC · 10.5 square units
.
square units: a trapezium with parallel sides 5 and 2 and height 3.
Worth noting
15 is the rectangle , as if every strip were as long as the bottom one.
What is the area of the region bounded by the line , the -axis and the lines and ?
AnswerD · 6.5 square units
The line crosses the -axis at : it is above the axis for and below it for .
square units.
Worth noting
is in one go: the part below the axis comes out negative and cancels 2 of the part above.
The line divides the region bounded by , the -axis and the line into two parts of equal area. What is ?
AnswerC ·
The whole region has area , so each part has 8.
, so .
Worth noting
2 halves the base, but the strips grow taller towards , so the line that halves the area lies to the right of the midpoint.
A region lies in the first quadrant. It is bounded by the -axis, the lines and , and the line .
The line cuts into two parts. What is the area of the part to the right of ?
AnswerA · 4.5 square units
For , , so every strip of crosses ; the part to the right has strips of length .
square units, which is .
Worth noting
6 is the part to the left of , the rectangle .
Using integration, what is the area of the triangle bounded by the lines , and ?
AnswerB · 6 square units
Corners: the first two lines meet at the origin; meets at and at .
The upper side changes at : .
square units.
Worth noting
3 stops at , where the upper side changes from to ; the second piece is left out.
Using integration, what is the area enclosed by the circle ?
AnswerD · square units
By symmetry, 4 times the quadrant: .
The quadrant is , so the area is square units.
Worth noting
is one quadrant; the circle is 4 of them.
Which integral gives the area of the part of the circle above the -axis?
AnswerD ·
Above the -axis the circle is , for .
Area .
Worth noting
covers only the first-quadrant quarter.
The circle is cut by the line into a smaller and a larger part.
At which points does the line meet the circle ?
AnswerD · and
, so .
.
Worth noting
stops at .
What is the area of the part of the circle in the first quadrant?
AnswerA · square units
with .
square units.
Worth noting
is the whole circle.
What is the area of the region bounded by the curve and the -axis?
AnswerC · square units
is the right half of the circle , from to .
Horizontal strips: square units, half the circle.
Worth noting
runs the strips from only and leaves out the half below the -axis.
Read the two statements and choose the correct option.
I. The area enclosed by is twice the area enclosed by .
II. The part of in the first quadrant has the same area as the whole circle .
AnswerC · Both I and II
I: the right sides are the squares of the radii, so the areas are and . True.
II: , the area of a circle of radius 1. True.
Worth noting
Doubting I reads 4 and 2 as the radii; they are the squares of the radii, so the areas are in the ratio .
For , the part of the circle in the first quadrant has area . What is ?
AnswerC ·
The quadrant is .
gives , so .
Worth noting
2 sets the whole circle, , equal to ; the area given is only a quarter of it.
Assertion (A): The area enclosed by the circle is square units.
Reason (R): The radius of the circle is 12.
AnswerC · A is true, but R is false.
The radius is , not 12, so R is false.
, so A is true.
Worth noting
A radius of 12 would give ; A holds because the right side 12 is the square of the radius.
Which integral gives the area enclosed by the circle ?
AnswerD ·
In the first quadrant the circle is , from to .
The four quadrants are equal, so the area is 4 times that integral, which is .
Worth noting
Without the factor 4 the integral is a single quadrant.
What is the area of the smaller region cut off from the circle by the line ?
AnswerA · square units
The line meets the circle at and : the region is the quarter circle minus the triangle under the line.
square units.
Worth noting
is the whole quarter circle; the triangle of area 2 has to come off.
The circle is cut by the line into a smaller and a larger part.
What is the area of the larger part of the circle, to the left of ?
AnswerB · square units
Larger part = whole circle − smaller part.
.
Worth noting
Adding the smaller part goes past the whole circle.
The circle is cut by the line into a smaller and a larger part.
Which integral gives the area of the smaller part, to the right of ?
AnswerD ·
The part runs from to , and the circle is symmetric about the -axis.
Upper half , doubled: .
Worth noting
Without the 2 the integral covers only the half above the -axis.
What is the area enclosed by the circle ?
AnswerD · square units
Complete the square: , a circle of radius 4 with centre .
Moving a region does not change its area: square units.
Worth noting
reads the constant 12 as the square of the radius before completing the square; it is .
The region inside the circle and above the line is found with vertical strips. Between which limits of is the integral taken?
AnswerD · to
The line meets the circle where , so .
Between them each strip runs from up to : .
Worth noting
are where the circle meets the -axis; above the region stops where the line meets the circle.
What is the area of the region bounded by the arc of in the first quadrant and the lines and ?
AnswerA · square units
Each vertical strip runs from the arc up to , for .
square units: the square minus the quarter circle.
Worth noting
is the quarter circle under the arc; the region asked for is the corner of the square above it.
Match each part of the circle in List I with its area in List II.
| List I | List II |
|---|---|
| A. The part above the -axis | I. |
| B. The part in the first quadrant | II. |
| C. The part in the first quadrant below the line | III. |
| D. The part to the right of the line | IV. |
AnswerD · A-II, B-IV, C-I, D-III
A is half of , B a quarter, and the line cuts the quadrant into two equal halves, so C is .
D: .
Worth noting
Swapping A and B confuses the half above the axis with the quarter in the first quadrant.
For the circle , which of the following statements are true?
I. Its radius is .
II. .
III. The area it encloses is .
IV. Its part in the second quadrant has area .
AnswerA · I, II and IV only
I: , so . II: the integral is the half above the -axis, .
III is false: the upper limit must be the radius , not 20. IV: each quadrant is .
Worth noting
Taking III as true puts the limit at 20, the right side of the equation, instead of the radius .
What is the area of the region inside the circle and outside the square ?
AnswerC · square units
In the first quadrant the side of the square is the chord ; the piece there is .
The four quadrants give equal pieces: square units.
Worth noting
is the piece in one quadrant; the square cuts off four equal pieces.
Which integral gives the area of the region in the first quadrant that lies below the line and inside the circle ?
AnswerB ·
The region lies against the -axis from to ; at height a strip runs from to the circle, .
So the area is .
Worth noting
runs vertical strips up to the arc, which rises above the line for .
The figure shows the smaller part of the circle cut off by the line . What is its area?
AnswerB · square units
The part is symmetric about the -axis: .
; between the limits this gives , so the area is .
Worth noting
is the part above the -axis only.
What is the area of the region in the first quadrant bounded by the -axis, the line and the circle ?
AnswerB · square units
The line meets the circle at . Split at : .
square units: an eighth of the circle, .
Worth noting
is the second integral alone; the triangle under adds 2.
What is the area of the region in the first quadrant between the circle and the line ?
AnswerA · square units
Quarter circle minus the triangle under the line: .
square units.
Worth noting
is the whole quarter circle, before taking away the triangle.
What is the area of the part of the circle to the right of the line ?
AnswerB · square units
, using .
Between the limits: ; doubled, .
Worth noting
is the half above the -axis only.
The circle is cut by the line into a smaller and a larger part.
What is the area of the smaller part of the circle, to the right of ?
AnswerC · square units
, using .
Between 3 and 5: ; doubled, .
Worth noting
Half of this is the part above the -axis only.
For , what is the area of the region in the first quadrant inside the circle and to the right of the line ?
AnswerD ·
.
At the bracket is ; at it is .
Difference: square units.
Worth noting
keeps only the terms and drops at .
What is the area of the part of the circle between the lines and ?
AnswerC · square units
The part is symmetric about both axes: .
.
Area square units.
Worth noting
is the part above the -axis only.
What is the area of the region ?
AnswerB · square units
meets the circle where , at ; the region is symmetric about the -axis.
Right half: .
Both halves: square units, a quarter of the disc.
Worth noting
is the half with ; the region has a mirror image for as well.
What is the area of the region in the first quadrant that lies below the line and inside the circle ?
AnswerB · square units
The region lies against the -axis: horizontal strips from to , for .
.
square units.
Worth noting
takes as ; it is .
What is the area of the region in the first quadrant bounded by , the -axis and the lines and ?
AnswerB · 14 square units
In the first quadrant : .
square units.
Worth noting
28 doubles the region, as if both halves of the parabola counted; the question asks for the first quadrant.
What is the area of the region between and ?
AnswerA · square units
They meet where : and , with the line above.
square units.
Worth noting
is the area under the line alone.
The parabola has its focus at , and its latus rectum is the chord .
Where does the latus rectum meet the parabola ?
AnswerD · and
.
.
Worth noting
stops at .
What is the area of the region in the first quadrant bounded by and the line ?
AnswerA · 18 square units
In the first quadrant : .
square units.
Worth noting
36 doubles the region, counting the part below the -axis too.
What is the area of the region bounded by , the -axis and the lines and ?
AnswerB · square units
.
square units.
Worth noting
uses only the upper limit, as if the region started at .
The parabola has its focus at , and its latus rectum is the chord .
What is the whole area bounded by and its latus rectum?
AnswerC · square units
Double the half above the axis: .
square units, which is with .
Worth noting
4 is the rectangle with sides 1 and 4 around the region.
The region bounded by the parabola and the line is found with vertical strips. Between which limits of is the integral taken?
AnswerA · to
The line meets the parabola where , so .
Between them the line lies above the parabola: .
Worth noting
solves , forgetting that on the right side is .
Using horizontal strips, which integral gives the area of the region bounded by the parabola and the line ?
AnswerD ·
gives , so a strip at height runs from to , length .
The strips run from the vertex, , to : .
Worth noting
measures each strip only from the -axis to the right arm, half the region.
What is the area of the region bounded by the parabola and the line ?
AnswerB · square units
The line meets the parabola where , at ; between them the parabola lies above the line .
square units, which is with .
Worth noting
is : the part of the rectangle between the parabola and the -axis, outside the region.
What is the area of the region in the first quadrant bounded by , the -axis and the lines and ?
AnswerC · square units
The region lies against the -axis, so use horizontal strips: , .
square units.
Worth noting
integrates along , the area under the curve, not beside it.
The figure shows the region bounded by the parabola and the line , its latus rectum. What is its area?
AnswerC · square units
The region is symmetric about the -axis: .
square units.
Worth noting
is the half above the -axis.
What is the area of the region bounded by the curve and the line ?
AnswerC · square units
They meet at ; the line lies above: .
square units.
Worth noting
is the area under the parabola from to 2; the region is the part above it, inside the line.
For which is the area under from to equal to 9 square units?
AnswerB ·
.
, so .
Worth noting
27 is ; the cube root is still to be taken.
What is the area of the region between the parabola and the line ?
AnswerA · square units
gives ; between them lies above .
square units.
Worth noting
is the area under the parabola's upper half alone.
What is the area of the region bounded by and the line ?
AnswerC · square units
By symmetry, .
square units.
Worth noting
is the half above the -axis.
What is the area of the region bounded by and the line ?
AnswerB · square units
They meet at ; the line lies above : .
square units.
Worth noting
is the area under the parabola, not the region above it.
The parabola has its focus at , and its latus rectum is the chord .
What is the area of the part of this region above the -axis?
AnswerB · square units
.
square units.
Worth noting
integrates , forgetting the factor 2 in .
Read the two statements and choose the correct option.
I. When a region lies below the -axis, its integral is negative and the area is the absolute value.
II. If a line cuts a curve and lies below it between the points where they meet, the region cut off has area between those points.
AnswerC · Both I and II
I: , area . True.
II: for example for over . True.
Read the two statements and choose the correct option.
I. For , the region bounded by and the line has of the area of the rectangle , .
II. The region bounded by and the line has twice the area of the region bounded by , the -axis and the lines and .
AnswerC · Both I and II
I: , and the rectangle is ; the ratio is . True.
II: and . True.
Worth noting
Doubting II mixes up the two regions: the one inside the parabola is what is left of the rectangle, .
What is the area of the region bounded by the curve and the -axis?
AnswerC · square units
: the curve meets the -axis at and and lies below it between them.
, so the area is square units.
Worth noting
is the integral; the region lies below the -axis, so the area is its absolute value.
Which expression gives the area of the region in the first quadrant bounded by the curve , the line and the -axis?
AnswerA ·
meets where , at in the first quadrant; the line meets the -axis at .
The top edge is the parabola for and the line for : .
Worth noting
is the region between the line and the parabola, which is closed off by the -axis, not the -axis.
For , what is the area of the region bounded by the parabola and the line ?
AnswerA ·
meets at and lies above it between them.
square units.
Worth noting
is , the area under the parabola, not the region above it.
Let be the area bounded by the parabola and the line , and the area bounded by the same parabola and the line . What is ?
AnswerC ·
Up to the area is : and .
.
Worth noting
4 assumes the area grows like ; it grows like , because the region also gets taller as it gets longer.
What is the area of the region bounded by the parabola and the line ?
AnswerC · 4.5 square units
gives and ; between them the line lies above the parabola.
square units.
Worth noting
is the area under the line alone; the 3 under the parabola has to come off.
For the region bounded by the parabola and the line , which of the following statements are true?
I. The line meets the parabola at and .
II. Its area is .
III. Its area is square units.
IV. Its area is .
AnswerD · I, II and III only
I: , so . II and III: the line lies above the parabola and the region is symmetric about the -axis, so the area is .
IV: a horizontal strip runs from to , so its length is ; is only half the area. False.
Worth noting
Taking IV as true counts only the right half of each horizontal strip.
The area of the region bounded by and its latus rectum is . For which is this area 24 square units?
AnswerB ·
gives .
.
Worth noting
9 is ; the square root is still to be taken.
The parabola has its focus at , and its latus rectum is the chord .
What is the area of the region of the parabola between the lines and ?
AnswerC · square units
.
square units.
Worth noting
runs from , counting the part up to the latus rectum as well.
What is the area of the region bounded by the parabola and the line ?
AnswerB · 9 square units
As in terms of : the parabola is and the line . They meet where , at and .
Horizontal strips run from the parabola to the line: .
square units.
Worth noting
uses vertical strips with the line as the lower edge all the way from ; for the lower edge is the parabola's lower arm.
What is the area of the region in the first quadrant bounded by the curve , the line and the -axis?
AnswerB · square units
The curve meets the line where , at ; the line meets the -axis at .
Horizontal strips run from the curve to the line : square units.
Worth noting
is the whole area under the curve up to ; the triangle of area 2 under the line is not part of the region.
For , the region bounded by the parabola and the line has area square units. What is ?
AnswerD ·
The line meets the parabola where , at .
Area .
gives , so .
Worth noting
2 is ; it still has to be squared.
The line divides the region bounded by the parabola and the line into two parts of equal area. What is ?
AnswerC ·
The part below is bounded by and , so its area is ; the whole region, with , has .
gives , so .
Worth noting
2 halves the height, but the region is wider at the top, so the line that halves the area lies above the middle.
For , the region bounded by the parabola and the line has area square units. What is ?
AnswerB ·
at and ; between them the line lies above the parabola.
.
gives , so .
Worth noting
8 stops at without taking the cube root.
Using integration, what is the area enclosed by the ellipse ?
AnswerB · square units
In the first quadrant ; by symmetry the area is .
square units, which is with , .
Worth noting
multiplies the denominators 25 and 16; the semi-axes are their square roots, 5 and 4.
What is the area of the part of the ellipse in the first quadrant?
AnswerA · square units
, so the quadrant is .
square units, a quarter of .
Worth noting
is the whole ellipse.
What is the area enclosed by the ellipse ?
AnswerA · square units
In standard form , so , .
Area square units.
Worth noting
reads as 4 without first dividing the equation by 16.
What is the area enclosed by the ellipse ?
AnswerA · square units
Divide by 36: , so and .
Area square units.
Worth noting
reads as the 36 on the right side, before putting the equation in standard form.
What is the area of the part of the ellipse that lies in the first quadrant?
AnswerD ·
.
, a quarter of the whole ellipse.
Worth noting
is the half above the -axis.
Which integral gives the area of the part of the ellipse above the -axis?
AnswerA ·
Above the -axis, , for .
So the area is , half of .
Worth noting
solves for with the semi-axes swapped; the 5 under goes on top.
For , what is the area enclosed by the ellipse ?
AnswerA ·
The semi-axes are and .
Area square units.
Worth noting
multiplies the denominators and instead of their square roots.
The ellipse meets the axes at and . Joining these four points in turn gives a quadrilateral inside the ellipse.
What is the area enclosed by the ellipse?
AnswerC · square units
and ; in the first quadrant .
Area square units, which is .
Worth noting
is the part in one quadrant; the ellipse is four such parts.
Read the two statements and choose the correct option.
I. The ellipse encloses the same area as a circle of radius .
II. The part of the ellipse above the -axis has area .
AnswerA · Only I
I: , the area of the ellipse. True.
II: the part above the -axis is half the ellipse, ; is one quadrant. False.
Worth noting
Taking II as true confuses the half above the axis with the quarter in one quadrant.
What is the area of the part of the ellipse to the right of the -axis?
AnswerB · square units
Divide by 36: , so , , and the ellipse encloses .
The -axis cuts it into two equal halves: square units.
Worth noting
is the whole ellipse; the part to the right of the -axis is half of it.
Which of the following statements are true?
I. The area of the ellipse is .
II. An ellipse with is a circle, and becomes .
III. The area of is .
IV. Doubling both semi-axes of an ellipse multiplies its area by 4.
AnswerD · I, II and IV only
I, II and IV hold: .
III is false: is not even in square units of length times length.
Which integral gives the area enclosed by the ellipse ?
AnswerD ·
Solving for in the first quadrant: , for .
Four equal quadrants give .
Worth noting
solves for with the semi-axes swapped.
The ellipse , , encloses an area of . What is ?
AnswerC ·
with .
: .
Worth noting
takes , the denominator, instead of its square root.
The ellipse meets the axes at and . Joining these four points in turn gives a quadrilateral inside the ellipse.
What is the area of ?
AnswerC · 48 square units
In the first quadrant the side of is , that is : .
is symmetric about both axes: square units, half of .
Worth noting
12 is the triangle in the first quadrant only; is four such triangles.
The ellipse meets the axes at and . Joining these four points in turn gives a quadrilateral inside the ellipse.
What is the area of the region inside the ellipse but outside ?
AnswerB · square units
Inside the ellipse and outside is the ellipse with taken away.
square units: four equal pieces of , one between each quarter of the ellipse and its chord.
Worth noting
is the piece in one quadrant; there are four.
The region inside the ellipse and above the line is found with vertical strips. Between which limits of is the integral taken?
AnswerD · to
On : , so and .
Between these the arc lies above the line: .
Worth noting
are where the ellipse meets the -axis; above the region is narrower and ends where the line meets the ellipse.
Assertion (A): The ellipse and the circle enclose equal areas.
Reason (R): The area enclosed by is .
AnswerA · Both A and R are true, and R is the correct explanation of A.
R: . True.
So the ellipse encloses , and the circle, the case , encloses : A is true, and R explains it.
Worth noting
Doubting A compares 16 and 9 with 12; the areas depend on and on the square of the radius, also 12.
For the ellipse , which of the following statements are true?
I. Its area is .
II. Its area is .
III. Its area is .
IV. Its part in the first quadrant has area .
AnswerD · II, III and IV only
Standard form: , so , and the area is , not . I is false, and IV, a quarter of , is true.
II solves for and III for ; each gives , so both are true.
Worth noting
Taking I as true reads the 36 on the right side as the product of the semi-axes; in standard form they are 2 and 3.
For , the ellipse encloses the same area as the circle . What is ?
AnswerB ·
The circle has radius , so it encloses ; the ellipse has and encloses .
gives .
Worth noting
takes , the denominator itself, instead of its square root 5.
Match each ellipse in List I with the area it encloses in List II.
| List I | List II |
|---|---|
| A. | I. |
| B. | II. |
| C. | III. |
| D. | IV. |
AnswerD · A-III, B-I, C-IV, D-II
A: , . B: dividing by 4, , so , .
C: , area . D: , area .
Worth noting
Swapping A and B misreads B: dividing by 4 gives the semi-axes 1 and 2.
What fraction of the area enclosed by the circle does the ellipse enclose?
AnswerD ·
The ellipse has , : it encloses ; the circle encloses .
, which is : each vertical strip of the ellipse is of the circle's strip.
Worth noting
compares the denominators 9 and 25; the strips shrink by , not by its square.
The figure shows the region in the first quadrant between the ellipse and the chord . What is its area?
AnswerA · square units
Quarter ellipse minus the triangle under the chord: .
square units.
Worth noting
is the whole quarter ellipse, before taking away the triangle of area 3.
For the ellipse , what is the area of the region in the first quadrant between and ?
AnswerA · square units
: .
square units.
Worth noting
halves only one of the two terms.
What is the area of the region inside the ellipse and above the line ?
AnswerB · square units
meets the ellipse where , at ; the upper arc is .
.
Take away the rectangle under , of area : square units.
Worth noting
is the area under the arc down to the -axis; the rectangle below still has to come off.
What is the area of the region in the first quadrant bounded by the -axis, the line and the ellipse ?
AnswerC · square units
The line meets the ellipse where , at .
Under the line up to , then under the arc: .
square units, an eighth of the ellipse's .
Worth noting
is the second integral alone; the triangle under the line adds 3.
The ellipse meets the axes at and . Joining these four points in turn gives a quadrilateral inside the ellipse.
What fraction of the area of the ellipse lies inside ?
AnswerD ·
.
For any ellipse, has area and the ellipse , so the fraction is always .
Worth noting
is the fraction outside .
For , what is the area of the smaller region bounded by the ellipse and the line ?
AnswerA ·
The line joins and , both on the ellipse, so the smaller region lies in the first quadrant between the arc and the chord.
.
square units.
Worth noting
is the whole quarter of the ellipse; the triangle under the chord, , has to come off.
What is the area of the region in the first quadrant that lies below the line and inside the ellipse ?
AnswerA · square units
The region lies against the -axis, so solve for : , for .
.
square units.
Worth noting
forgets the factor in .
NTA's Mathematics / Applied Mathematics syllabus has three sections: Section A1, which every candidate answers, then Section B1 (Mathematics) or Section B2 (Applied Mathematics), of which a candidate answers one. Each section lists units and topics but no chapters, so the chapters here are ours. Where both B sections list a topic that Section A also lists, its questions sit once, in Section A.