The functions and intersect at and

MATHEMATICS · Single correct · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

The functions p(x)=sin⁡xp(x)=\sin x and q(x)=cos⁡xq(x)=\cos x intersect at x=π4x=\tfrac{\pi}{4} and x=5π4x=\tfrac{5\pi}{4}. A student sets up the integral ∫π/45π/4(sin⁡x−cos⁡x) dx\displaystyle\int_{\pi/4}^{5\pi/4}(\sin x-\cos x)\,dx to find the area of the region bounded by the two graphs on that interval. Which statement best evaluates the student’s setup? (a) It is invalid because the region switches which curve is on top halfway through, so the interval must be divided at x=πx=\pi into two separate integrals. (b) It is invalid because cos⁡x\cos x exceeds sin⁡x\sin x near x=π2x=\tfrac{\pi}{2}, so the integrand should be cos⁡x−sin⁡x\cos x-\sin x instead. (c) It is valid because sin⁡x≥cos⁡x\sin x\ge \cos x throughout the interval, so the integrand is non-negative. (d) It is valid only if the absolute value of sin⁡x−cos⁡x\sin x-\cos x is integrated; otherwise the negative part of the difference will be subtracted.

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