Difference between revisions of "009B Sample Final 1, Problem 5"

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!Step 1:    
 
!Step 1:    
 
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|We need to integrate
 
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|<math>\int_0^1 2\pi x(e^x-ex)~dx=2\pi\int_0^1 xe^x~dx-2\pi\int_0^1ex^2~dx</math>.
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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|For the first integral, we need to use integration by parts.
 
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|Let <math>u=x</math> and <math>dv=e^xdx</math>. Then, <math>du=dx</math> and <math>v=e^x</math>.
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|So, the integral becomes
 
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::<math>\begin{array}{rcl}
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\displaystyle{\int_0^1 2\pi x(e^x-ex)~dx} & = & \displaystyle{2\pi\bigg(xe^x\bigg|_0^1 -\int_0^1 e^xdx\bigg)-\frac{2\pi ex^3}{3}\bigg|_0^1}\\
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&&\\
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& = & \displaystyle{2\pi\bigg(xe^x-e^x\bigg)\bigg|_0^1-\frac{2\pi e}{3}}\\
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&&\\
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& = & \displaystyle{2\pi(e-e-(-1))-\frac{2\pi e}{3}}\\
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&&\\
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& = & \displaystyle{2\pi-\frac{2\pi e}{3}}\\
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\end{array}</math>
 
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|'''(b)''' <math>\int_0^1 2\pi x(e^x-ex)~dx</math>
 
|'''(b)''' <math>\int_0^1 2\pi x(e^x-ex)~dx</math>
 
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|'''(c)'''  
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|'''(c)''' <math>2\pi-\frac{2\pi e}{3}</math>
 
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[[009B_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]
 
[[009B_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]

Revision as of 18:55, 4 February 2016

Consider the solid obtained by rotating the area bounded by the following three functions about 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 y} -axis:

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} , 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=e^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=ex} .

a) Sketch the region bounded by the given three functions. Find the intersection point of the two 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=e^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=ex} . (There is only one.)

b) Set up the integral for the volume of the solid.

c) Find the volume of the solid by computing the integral.

Foundations:  
Review volumes of revolutions

Solution:

(a)

Step 1:  
First, we sketch the region bounded by the three functions.
Insert graph here.
Step 2:  
Setting the equations equal, we have 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 e^x=ex} .
We get one intersection point, which 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 (1,e)} .
This intersection point can be seen in the graph shown in Step 1.

(b)

Step 1:  
We proceed using cylindrical shells. The radius of the shells 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=x} .
The height of the shells 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 h=e^x-ex} .
Step 2:  
So, 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 \int_0^1 2\pi x(e^x-ex)~dx} .

(c)

Step 1:  
We need to integrate
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_0^1 2\pi x(e^x-ex)~dx=2\pi\int_0^1 xe^x~dx-2\pi\int_0^1ex^2~dx} .
Step 2:  
For the first integral, we need to use integration by parts.
Let 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 u=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 dv=e^xdx} . Then, 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 du=dx} 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 v=e^x} .
So, the integral becomes
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{\int_0^1 2\pi x(e^x-ex)~dx} & = & \displaystyle{2\pi\bigg(xe^x\bigg|_0^1 -\int_0^1 e^xdx\bigg)-\frac{2\pi ex^3}{3}\bigg|_0^1}\\ &&\\ & = & \displaystyle{2\pi\bigg(xe^x-e^x\bigg)\bigg|_0^1-\frac{2\pi e}{3}}\\ &&\\ & = & \displaystyle{2\pi(e-e-(-1))-\frac{2\pi e}{3}}\\ &&\\ & = & \displaystyle{2\pi-\frac{2\pi e}{3}}\\ \end{array}}
Final Answer:  
(a) 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 (1,e)} (See (a) Step 1 for the graph)
(b) 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_0^1 2\pi x(e^x-ex)~dx}
(c) 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 2\pi-\frac{2\pi e}{3}}

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