The region bounded above by

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

Problem

The region bounded above by y=sin⁡xy = \sin x, below by the xx-axis, and between x=0x = 0 and x=πx = \pi is revolved about the horizontal line y=2y = 2. Which of the following integrals represents the volume of the resulting solid? (a) ∫0π2πsin⁡x (2−sin⁡x) dx\displaystyle \int_{0}^{\pi} 2\pi\sin x\,(2-\sin x)\,dx (b) ∫0ππ[4−sin⁡2x] dx\displaystyle \int_{0}^{\pi} \pi\bigl[4-\sin^{2}x\bigr]\,dx (c) ∫0ππ[4−(2−sin⁡x)2] dx\displaystyle \int_{0}^{\pi} \pi\bigl[4-(2-\sin x)^{2}\bigr]\,dx (d) ∫0ππ(2−sin⁡x)2 dx\displaystyle \int_{0}^{\pi} \pi(2-\sin x)^{2}\,dx

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