Difference between revisions of "009B Sample Final 1, Problem 5"
Jump to navigation
Jump to search
Kayla Murray (talk | contribs) |
Kayla Murray (talk | contribs) |
||
| Line 44: | Line 44: | ||
!Step 1: | !Step 1: | ||
|- | |- | ||
| − | | | + | |We proceed using cylindrical shells. The radius of the shells is given by <math>r=x</math>. |
| − | |||
| − | |||
|- | |- | ||
| − | | | + | |The height of the shells is given by <math>h=e^x-ex</math>. |
|} | |} | ||
| Line 54: | Line 52: | ||
!Step 2: | !Step 2: | ||
|- | |- | ||
| − | | | + | |So, the volume of the solid is <math>\int_0^1 2\pi x(e^x-ex)~dx</math>. |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
|} | |} | ||
| Line 90: | Line 80: | ||
|'''(a)''' <math>(1,e)</math> (See (a) Step 1 for the graph) | |'''(a)''' <math>(1,e)</math> (See (a) Step 1 for the graph) | ||
|- | |- | ||
| − | |'''(b)''' | + | |'''(b)''' <math>\int_0^1 2\pi x(e^x-ex)~dx</math> |
|- | |- | ||
|'''(c)''' | |'''(c)''' | ||
|} | |} | ||
[[009B_Sample_Final_1|'''<u>Return to Sample Exam</u>''']] | [[009B_Sample_Final_1|'''<u>Return to Sample Exam</u>''']] | ||
Revision as of 18:35, 4 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 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: |
|---|
| Step 2: |
|---|
| 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) |