Single Variable Calculus

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Single Variable Calculus, Chapter 8, 8.2, Section 8.2, Problem 62

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Find the volume obtained by rotating the region bounded by $y = \sin^2 x$ and $y = 0$ from $0 \leq x \leq \pi$ about $x$-axis.

By using vertical strips, if you slice the figure, you'll get a cross sections of circle with radius $r = y_{\text{upper}} - y_{\text{lower}} = \sin^x - 0 = \sin^2 x$. So, the cross sectional area is computed by $A = \pi r^2 = \pi \left( \sin^2 x \right)^2$. Thus, the volume is...

$ \begin{equation} \begin{aligned} V = \int^b_a A (x) dx &= \int^\pi_0 \pi \left( \sin^2 x \right)^2 && \text{ recall that } \sin^2 x = \frac{1 - \cos (2x)}{2}\\ \\ &= \pi \int^\pi_0 \left[ \frac{1 - \cos (2x) }{2} \right]^2 dx\\ \\ &= \frac{\pi}{4} \int^\pi_0 \left[ 1 - 2 \cos (2x) + \cos^2 (2x) \right] dx && \text{ recall that } \cos^2(x) = \frac{1 + \cos (2x) }{2}\\ \\ &= \frac{\pi}{4} \int^\pi_0 \left[ 1 - 2 \cos (2x) + \left[ \frac{1+\cos(2(2x))}{2} \right]\right]dx \\ \\ &= \frac{\pi}{4} \int^\pi_0 \left[ \frac{3}{2} - 2 \cos (2x) + \frac{\cos(4x)}{2} \right] dx\\ \\ &= \frac{\pi}{4} \left[ \int^\pi_0 \frac{3}{2}dx - 2 \int^\pi_0 \cos (2x) dx + \frac{1}{2} \int^\pi_0 \cos(4x) dx \right]\\ \\ &= \frac{\pi}{4} \left[ \left( \frac{3}{2} x \right)^\pi_0 - 2 \left( \frac{1}{2} \right) \left[ \sin (2x) \right]^\pi_0 + \frac{1}{2} \left( \frac{1}{4} \right) \left[ \sin (4x) \right]^\pi_0 \right]\\ \\ &= \frac{3 \pi^2}{8} \text{ cubic units} \end{aligned} \end{equation} $

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