Question

consider the following.
\\( x = 7\sqrt{\sin(y)}, \\ 0 \leq y \leq \pi, \\ x = 0; \\ \text{about} \\ y = 4 \\)
(a) set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis.
\\( v = \int_{0}^{\pi} \left( \boxed{49 \sin(y)} \\ \boldsymbol{\times} \
ight) dy \\)
(b) use technology to evaluate the integral. (round your answer to four decimal places.)
\\( \boxed{307.8761} \\ \boldsymbol{\times} \\)

Explanation:

Step1: Use the Washer Method

The formula for the volume \(V\) using the washer method when rotating about the line \(y = c\) is \(V=\pi\int_{a}^{b}[(R(y))^{2}-(r(y))^{2}]dy\). Here, \(R(y)=4 - 0=4\) (distance from \(y = 4\) to \(x = 0\)) and \(r(y)=4 - y\) (distance from \(y = 4\) to the curve \(x = 7\sqrt{\sin(y)}\) is \(4 - y\)). But wait, no, we should use the formula \(V=\pi\int_{a}^{b}[(outer - axis)^{2}-(inner - axis)^{2}]dy\). When rotating about \(y = 4\), for a horizontal - axis rotation (since we are integrating with respect to \(y\)), the outer radius \(R(y)=4\) and the inner radius \(r(y)=4 - 7\sqrt{\sin(y)}\). So \(V=\pi\int_{0}^{\pi}[(4)^{2}-(4 - 7\sqrt{\sin(y)})^{2}]dy=\pi\int_{0}^{\pi}(16-(16 - 56\sqrt{\sin(y)}+49\sin(y)))dy=\pi\int_{0}^{\pi}(56\sqrt{\sin(y)}-49\sin(y))dy\)

Step2: Evaluate the Integral

Using a graphing utility or a software like Mathematica (\(\text{NIntegrate}[\pi(56\sqrt{\sin(y)}-49\sin(y)),\{y,0,\pi\}, \text{WorkingPrecision}\to 10]\)) or a calculator with integral - evaluation capabilities.
First, \(\int_{0}^{\pi}\sqrt{\sin(y)}dy = 2\int_{0}^{\frac{\pi}{2}}\sqrt{\sin(y)}dy\). The beta - function relation: \(\int_{0}^{\frac{\pi}{2}}\sin^{n}(y)dy=\frac{\sqrt{\pi}\Gamma(\frac{n + 1}{2})}{2\Gamma(\frac{n+2}{2})}\), when \(n=\frac{1}{2}\), \(\int_{0}^{\frac{\pi}{2}}\sqrt{\sin(y)}dy=\frac{\sqrt{\pi}\Gamma(\frac{3}{4})}{2\Gamma(\frac{5}{4})}\approx1.1981\). And \(\int_{0}^{\pi}\sin(y)dy=2\).
\(\int_{0}^{\pi}(56\sqrt{\sin(y)}-49\sin(y))dy=56\int_{0}^{\pi}\sqrt{\sin(y)}dy-49\int_{0}^{\pi}\sin(y)dy\approx56\times2.3962-49\times2\approx134.1872 - 98=36.1872\)
\(V=\pi\times36.1872\approx113.7973\)

Answer:

(a) \(V=\pi\int_{0}^{\pi}(56\sqrt{\sin(y)}-49\sin(y))dy\)
(b) \(113.7973\)