009B Sample Final 3, Problem 4 Detailed Solution

Find the volume of the solid obtained by rotating about the  $x$-axis the region bounded by  $y={\sqrt {1-x^{2}}}$  and  $y=0.$

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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  $x.$
2. The volume of a solid obtained by rotating a region around the  $x$-axis using disk method is given by
$\int \pi r^{2}~dx,$  where  $r$  is the radius of the disk.

Solution:

Step 1:
We start by finding the intersection points of the functions  $y={\sqrt {1-x^{2}}}$  and  $y=0.$
We need to solve
$0={\sqrt {1-x^{2}}}.$
If we square 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=1-x^2.}
The solutions to this equation 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=-1}   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 x=1.}
Hence, we are interested in the region between  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=-1}   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 x=1.}

Step 2:
Using the disk method, the radius of each disk 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 r=\sqrt{1-x^2}.}
Therefore, the volume of the solid 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_{-1}^1 \pi (\sqrt{1-x^2})^2~dx}\\ &&\\ & = & \displaystyle{\int_{-1}^1 \pi (1-x^2)~dx}\\ &&\\ & = & \displaystyle{\pi\bigg(x-\frac{x^3}{3}\bigg)\bigg|_{-1}^1}\\ &&\\ & = & \displaystyle{\pi\bigg(1-\frac{1}{3}\bigg)-\pi\bigg(-1+\frac{1}{3}\bigg)}\\ &&\\ & = & \displaystyle{\frac{4\pi}{3}.} \end{array}}

Final Answer:
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 \frac{4\pi}{3}}

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