Difference between revisions of "009B Sample Final 2, Problem 3"

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!Foundations:    
 
!Foundations:    
 
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|'''1.''' You can find the intersection points of two functions, say &nbsp; <math style="vertical-align: -5px">f(x),g(x),</math>
 
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&nbsp; &nbsp; &nbsp; &nbsp; by setting &nbsp;<math style="vertical-align: -5px">f(x)=g(x)</math>&nbsp; and solving for &nbsp;<math style="vertical-align: 0px">x.</math>
 
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|'''2.''' The volume of a solid obtained by rotating an area around the &nbsp;<math style="vertical-align: -4px">x</math>-axis using the washer method is given by 
 
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&nbsp; &nbsp; &nbsp; &nbsp; <math style="vertical-align: -13px">\int \pi(r_{\text{outer}}^2-r_{\text{inner}}^2)~dx,</math>&nbsp; where &nbsp;<math style="vertical-align: 0px">r_{\text{inner}}</math>&nbsp; is the inner radius of the washer and &nbsp;<math style="vertical-align: 0px">r_{\text{outer}}</math>&nbsp; is the outer radius of the washer.
 
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
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|First, we need to find the intersection points of &nbsp;<math>y=x</math>&nbsp; and &nbsp;<math>y=x^2.</math>
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|To do this, we need to solve &nbsp;<math>x=x^2.</math>
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|Moving all the terms on one side of the equation, we get
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
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\displaystyle{0} & = & \displaystyle{x^2-x}\\
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&&\\
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& = & \displaystyle{x(x-1).}
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\end{array}</math>
 
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|Hence, these two curves intersect at &nbsp;<math>x=0</math>&nbsp; and &nbsp;<math>x=1.</math>
 
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|So, we are interested in the region between &nbsp;<math>x=0</math>&nbsp; and &nbsp;<math>x=1.</math>
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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|We use the washer method to calculate this volume.
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|The outer radius is &nbsp;<math>r_{\text{outer}}=2-x^2</math>&nbsp; and
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|the inner radius is &nbsp;<math>r_{\text{inner}}=2-x.</math>&nbsp;
 
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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_0^1 \pi(r_{\text{outer}}^2-r_{\text{inner}}^2)~dx}\\
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&&\\
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& = & \displaystyle{\int_0^1 \pi((2-x^2)^2-(2-x)^2)~dx.}
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\end{array}</math>
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|}
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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!Step 3: &nbsp;
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|Now, we integrate to get
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
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\displaystyle{V} & = & \displaystyle{\pi \int_0^1 ((4-4x^2+x^4)-(4-4x+x^2))~dx}\\
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&&\\
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& = & \displaystyle{\pi \int_0^1 (4x-5x^2+x^4)~dx}\\
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&&\\
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& = & \displaystyle{\pi\bigg(2x^2-\frac{5x^3}{3}+\frac{x^5}{5}\bigg)\bigg|_0^1}\\
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&&\\
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& = & \displaystyle{\pi\bigg(2-\frac{5}{3}+\frac{1}{5}\bigg)-0}\\
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&&\\
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& = & \displaystyle{\frac{8\pi}{15}.}
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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{8\pi}{15}</math>
 
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[[009B_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]
 
[[009B_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

Revision as of 16:13, 4 March 2017

Find the volume of the solid obtained by rotating the region bounded by the curves    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=x^2}   about the line  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=2.}

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 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=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=x^2.}
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 x=x^2.}
Moving all the terms on one side of the equation, 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 \begin{array}{rcl} \displaystyle{0} & = & \displaystyle{x^2-x}\\ &&\\ & = & \displaystyle{x(x-1).} \end{array}}
Hence, these two curves intersect at  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}   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.}
So, 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=0}   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:  
We use the washer method to calculate this volume.
The outer radius 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 r_{\text{outer}}=2-x^2}   and
the inner radius 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 r_{\text{inner}}=2-x.}  
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_0^1 \pi(r_{\text{outer}}^2-r_{\text{inner}}^2)~dx}\\ &&\\ & = & \displaystyle{\int_0^1 \pi((2-x^2)^2-(2-x)^2)~dx.} \end{array}}
Step 3:  
Now, we integrate to 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 \begin{array}{rcl} \displaystyle{V} & = & \displaystyle{\pi \int_0^1 ((4-4x^2+x^4)-(4-4x+x^2))~dx}\\ &&\\ & = & \displaystyle{\pi \int_0^1 (4x-5x^2+x^4)~dx}\\ &&\\ & = & \displaystyle{\pi\bigg(2x^2-\frac{5x^3}{3}+\frac{x^5}{5}\bigg)\bigg|_0^1}\\ &&\\ & = & \displaystyle{\pi\bigg(2-\frac{5}{3}+\frac{1}{5}\bigg)-0}\\ &&\\ & = & \displaystyle{\frac{8\pi}{15}.} \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{8\pi}{15}}

Return to Sample Exam

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