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

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!Step 1:    
 
!Step 1:    
 
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|-
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|We proceed using cylindrical shells. The radius of the shells is given by <math>r=x</math>.
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|We proceed using cylindrical shells. The radius of the shells is given by <math style="vertical-align: 0px">r=x</math>.
 
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|-
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|The height of the shells is given by <math>h=e^x-ex</math>.
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|The height of the shells is given by <math style="vertical-align: -1px">h=e^x-ex</math>.
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
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|So, the volume of the solid is <math>\int_0^1 2\pi x(e^x-ex)~dx</math>.
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|So, the volume of the solid is  
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|
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::<math>\int_0^1 2\pi x(e^x-ex)~dx</math>.
 
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|We need to integrate  
 
|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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|
 +
::<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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|For the first integral, we need to use integration by parts.  
 
|For the first integral, we need to use integration by parts.  
 
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|-
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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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|Let <math style="vertical-align: 0px">u=x</math> and <math style="vertical-align: 0px">dv=e^xdx</math>. Then, <math style="vertical-align: 0px">du=dx</math> and <math style="vertical-align: 0px">v=e^x</math>.  
 
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|-
 
|So, the integral becomes
 
|So, the integral becomes
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
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|-
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|'''(a)''' <math>(1,e)</math> (See (a) Step 1 for the graph)
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|'''(a)''' <math style="vertical-align: -5px">(1,e)</math> (See Step 1 for the graph)
 
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|-
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|'''(b)''' <math>\int_0^1 2\pi x(e^x-ex)~dx</math>
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|'''(b)''' <math style="vertical-align: -5px">\int_0^1 2\pi x(e^x-ex)~dx</math>
 
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|'''(c)''' <math>2\pi-\frac{2\pi e}{3}</math>
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|'''(c)''' <math style="vertical-align: -14px">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 11:38, 22 February 2016

Consider the solid obtained by rotating the area bounded by the following three functions about the -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 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 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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