Which of the following statements are true? I. ∫ₐ^b f(x) dx = ∫ₐ^b f(t) dt. II. ∫ₐ^a f(x) dx = 0. III. ∫₀^2a f(x) dx = 2∫₀^a f(x) dx for every f. IV. ∫ₐ^b f(x) dx = -∫_b^a f(x) dx.

Question 47

MediumOriginal question

Which of the following statements are true?
I. $\int_a^b f(x)\,dx = \int_a^b f(t)\,dt$ .
II. $\int_a^a f(x)\,dx = 0$ .
III. $\int_0^{2a} f(x)\,dx = 2\int_0^a f(x)\,dx$ for every $f$ .
IV. $\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$ .

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

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CorrectNot quiteAnswer D

Correct answer: D

I: the name of the variable does not matter. II: an interval of length 0 has no area. IV: reversing the limits changes the sign.
III holds only when $f(2a - x) = f(x)$ ; for $f(x) = x$ on $[0, 2]$ , $\int_0^2 x\,dx = 2$ but $2\int_0^1 x\,dx = 1$ .

Concept: Properties of definite integrals
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The rule this testsProperties of definite integrals

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