Difference between revisions of "009B Sample Final 3, Problem 4"

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&nbsp; &nbsp; &nbsp; &nbsp; <math style="vertical-align: -13px">\int \pi r^2~dx,</math>&nbsp; where &nbsp;<math style="vertical-align: 0px">r</math>&nbsp; is the radius of the disk.
 
&nbsp; &nbsp; &nbsp; &nbsp; <math style="vertical-align: -13px">\int \pi r^2~dx,</math>&nbsp; where &nbsp;<math style="vertical-align: 0px">r</math>&nbsp; is the radius of the disk.
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
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|We start by finding the intersection points of the functions <math>y=\sqrt{1-x^2}</math> and <math>y=0.</math>
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|We need to solve <math>0=\sqrt{1-x^2}.</math>
 
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|If we square both sides, we get
 
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|<math>0=1-x^2.</math>
 
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|The solutions to this equation are <math>x=-1</math> and <math>x=1.</math>
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|Hence, we are interested in the region between <math>x=-1</math> and <math>x=1.</math>
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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|Using the disk method, the radius of each disk is given by <math>r=\sqrt{1-x^2}.</math>
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|Therefore, the volume of the solid is
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
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\displaystyle{V} & = & \displaystyle{\int_{-1}^1 \pi (\sqrt{1-x^2})^2~dx}\\
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&&\\
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& = & \displaystyle{\int_{-1}^1 \pi (1-x^2)~dx}\\
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&&\\
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& = & \displaystyle{\pi\bigg(x-\frac{x^3}{3}\bigg)\bigg|_{-1}^1}\\
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&&\\
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& = & \displaystyle{\pi\bigg(1-\frac{1}{3}\bigg)-\pi\bigg(-1+\frac{1}{3}\bigg)}\\
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&&\\
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& = & \displaystyle{\frac{4\pi}{3}.}
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\end{array}</math>
 
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>\frac{4\pi}{3}</math>
 
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[[009B_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]
 
[[009B_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

Revision as of 11:23, 3 March 2017

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

Foundations:  
1. You can find the intersection points of two functions, say   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 f(x),g(x),}

        by setting  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 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 a region 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 disk 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^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}   is the radius of the disk.


Solution:

Step 1:  
We start by finding the intersection points of the functions 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{1-x^2}} 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.}
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{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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