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Area of Bounded Regions · Question 109 of 115

The figure shows the region in the first quadrant between the ellipse x^2/9 + y^2/4 = 1 and the chord x/3 +…

Question 109

HardOriginal question

The figure shows the region in the first quadrant between the ellipse x29+y24=1\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1 and the chord x3+y2=1\frac{x}{3} + \frac{y}{2} = 1. What is its area?

An ellipse with semi-axes 3 and 2, and the chord from (3, 0) to (0, 2); the region between them in the first quadrant is shaded
Options for question 109

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CorrectNot quiteAnswer A

Correct answer: A

Explanation

  1. Quarter ellipse minus the triangle under the chord: ∫03(239−x2−(2−2x3))dx\int_0^{3} \left(\frac{2}{3}\sqrt{9 - x^{2}} - \left(2 - \frac{2x}{3}\right)\right)dx.

  2. 3π2−12×3×2=3π2−3\frac{3\pi}{2} - \frac{1}{2} \times 3 \times 2 = \frac{3\pi}{2} - 3 square units.

Worth noting

3π2\frac{3\pi}{2} is the whole quarter ellipse, before taking away the triangle of area 3.

Concept
An ellipse cut by a chord
Typical time
120 s
Your time
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The rule this testsEllipses
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