A curve is defined by and for

MATHEMATICS · Single correct · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

A curve is defined by x=etcos⁡tx=e^{t}\cos t and y=etsin⁡ty=e^{t}\sin t for 0≤t≤10\le t\le 1. Student A writes the arc-length integral IA=∫01(etcos⁡t−etsin⁡t)2+(etsin⁡t+etcos⁡t)2 dt.I_A=\displaystyle \int_{0}^{1} \sqrt{(e^{t}\cos t - e^{t}\sin t)^{2} + (e^{t}\sin t + e^{t}\cos t)^{2}}\,dt. Student B writes IB=∫01(etcos⁡t)2+(etsin⁡t)2 dt.I_B=\displaystyle \int_{0}^{1} \sqrt{(e^{t}\cos t)^{2} + (e^{t}\sin t)^{2}}\,dt. Which statement correctly compares the two integrals? (a) Both IAI_A and IBI_B are correct because, after using cos⁡2t+sin⁡2t=1\cos^{2}t+\sin^{2}t=1, they yield the same integrand. (b) Only IAI_A is correct; IBI_B omits terms that arise from differentiating cos⁡t\cos t and sin⁡t\sin t. (c) Only IBI_B is correct because the derivative terms are redundant once the expressions are squared inside the radical. (d) Neither integral is correct since the integrand should simplify to 2 et\sqrt{2}\,e^{t}, not an expression containing trigonometric terms.

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