Which of the following statements are true? I. A definite integral ∫ₐ^b f(x) dx is a number, not a function of x. II. ∫ₐ^a f(x) dx = 0 for any f defined at a. III. The constant C of the antiderivative changes the value of F(b) - F(a). IV. ∫ₐ^b f(x) dx = ∫ₐ^b f(t) dt.

Question 10

MediumOriginal question

Which of the following statements are true?
I. A definite integral $\int_a^b f(x)\,dx$ is a number, not a function of $x$ .
II. $\int_a^a f(x)\,dx = 0$ for any $f$ defined at $a$ .
III. The constant $C$ of the antiderivative changes the value of $F(b) - F(a)$ .
IV. $\int_a^b f(x)\,dx = \int_a^b f(t)\,dt$ .

I, II and IV onlyCorrect answerYour answerAll four: I, II, III and IVCorrect answerYour answerI and II onlyCorrect answerYour answerII, III and IV onlyCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

The limits are numbers, so the result is a number (I); with equal limits $F(a) - F(a) = 0$ (II); and the letter used for the variable does not matter (IV).
III is false: $(F(b) + C) - (F(a) + C) = F(b) - F(a)$ , so $C$ cancels.

Concept: What a definite integral is
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsThe fundamental theorem of calculus

Practise this chapter →

Ten questions from Definite Integrals & Area, at your own pace.

Explanation