007B Sample Midterm 3, Problem 4 Detailed Solution

Find the volume of the solid obtained by rotating the region bounded by  $y={\sqrt {\sin x}},~0\leq x\leq \pi ,$  and  $y=0$  about the  $x-$axis. Sketch the graph of the region and a typical disk element.

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Background Information:
1. You can find the intersection points of two functions, say   $f(x),g(x),$
by setting  $f(x)=g(x)$  and solving for  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x.}
2. The volume of a solid obtained by rotating an area around the  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -axis using the washer method is given by
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int \pi(r_{\text{outer}}^2-r_{\text{inner}}^2)~dx,}
where  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_{\text{inner}}}   is the inner radius of the washer and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_{\text{outer}}}   is the outer radius of the washer.

Solution:

Step 1:
First, we need to find the intersection points of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=\sqrt{\sin x}}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=0.}
To do this, we need to solve
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0=\sqrt{\sin x}.}
Squaring both sides, we get  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0=\sin x.}
The solutions to this equation in the interval  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0,\pi]}   are
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=0,\pi.}
Now, the graph of the region is below.
Additionally, we are going to be using the washer/disk method.
Below, we show a typically disk element.
(Insert graph)

Step 2:
The volume of the solid using the disk method is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{V} & = & \displaystyle{\int_0^\pi \pi(\sqrt{\sin x})^2~dx}\\ &&\\ & = & \displaystyle{\int_0^\pi \pi\sin x~dx}\\ &&\\ & = & \displaystyle{-\pi \cos x\bigg|_0^\pi }\\ &&\\ & = & \displaystyle{-\pi \cos(\pi)+\pi\cos(0)}\\ &&\\ & = & \displaystyle{2\pi.} \end{array}}

Final Answer:
See Step 1 for graph.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V=2\pi}

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