A student states: “Because

MATHEMATICS · Single correct · A Level · Mathematics · Integration · Integration

Problem

A student states: “Because ∫26p(x) dx=10\displaystyle \int_{2}^{6} p(x)\,dx = 10, it follows that ∫62p(x) dx=10\displaystyle \int_{6}^{2} p(x)\,dx = 10 since the same region is involved.” Which of the following best evaluates the student’s claim? (a) The claim is correct because the integral represents the same region, so its value is still 1010. (b) The claim is incorrect because reversing the limits makes the integral from 6 to 2 and 2 to 6 cancel, giving 00. (c) The claim is incorrect because reversing the limits multiplies the integral by −1-1, so ∫62p(x) dx=−10\displaystyle \int_{6}^{2} p(x)\,dx = -10. (d) The claim cannot be evaluated without knowing whether p(x)p(x) is positive on [2,6][2,6], because the sign of the integral could change.

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