Difference between revisions of "Prototype Calculus Question"

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[[File:9BSF1 3a.png|250px|right]]
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[[File:9BSF1 3a.png|270px|right]]
  
 
Find the volume of the solid obtained by rotating the area enclosed by <math> y=5-x </math> and <math style="vertical-align:-17%;"> y=25-x^2 </math> <br> around the ''x''-axis.
 
Find the volume of the solid obtained by rotating the area enclosed by <math> y=5-x </math> and <math style="vertical-align:-17%;"> y=25-x^2 </math> <br> around the ''x''-axis.
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Foundations
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! Foundations &nbsp;
 
|-
 
|-
|Choose either shell or washer method.
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|&bull; Choose either shell or washer method.
 
|-
 
|-
|Find the appropriate radii.
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|&bull; Find the appropriate radii.
 
|-
 
|-
|Determine the bounds of integration by finding when both functions have the same ''y'' value.  
+
|&bull; Determine the bounds of integration by finding when both functions have the same ''y'' value.  
 
|-
 
|-
|Using the determined values, set up and solve the integral.
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|&bull; Using the determined values, set up and solve the integral.
 
|}
 
|}
  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:
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!Step 1: &nbsp;
 
|-
 
|-
|Since we are rotating around the ''x''-axis, the washer method would utilize tall rectangles with ''dx'' as their width.  This seems like a reasonable choice, as these rectangles would be trapped between our two functions, allowing us to solve a single integral.
+
|'''Choosing the Approach:''' &nbsp; Since we are rotating around the ''x''-axis, the washer method would utilize tall rectangles with ''dx'' as their width.  This seems like a reasonable choice, as these rectangles would be trapped between our two functions as ''x'' varies over the enclosed region, allowing us to solve a single integral. Note that the washer method will require an inner and outer radius, as well as bounds of integration, in order to evaluate the integral
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|-
 +
|-
 +
|&nbsp;&nbsp;<math>V=\pi \int_{x_1}^{\, x_2} R^2-r^2 \,dx</math>
 
|}
 
|}
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:
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!Step 2: &nbsp;
 
|-
 
|-
|Since our rectangles will be trapped between the two functions, and will be rotated around the ''x''-axis (where <math> y=0 </math>,) we find
+
|'''Finding the Radii:''' &nbsp;Since our rectangles will be trapped between the two functions, and will be rotated around the ''x''-axis (where ''y'' = 0), we find
 
|-
 
|-
 
|the inner radius is <math style="vertical-align:0%;"> r = 5-x </math>, represented by the blue line, while
 
|the inner radius is <math style="vertical-align:0%;"> r = 5-x </math>, represented by the blue line, while
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 3:
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!Step 3: &nbsp;
 +
|-
 +
|'''Finding the Bounds of Integration:''' &nbsp; We must set the two functions equal, and solve.  If
 +
|-
 +
|<math>5-x=25-x^2,</math>
 +
|-
 +
|then by moving all terms to the left hand side and factoring,
 +
|-
 +
|<math>x^2-x-20=(x+4)(x-5)=0,</math>
 +
|-
 +
|so we have -4 and 5 as solutions.  These are our bounds of integration.
 +
|}
 +
 
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Step 4: &nbsp;
 +
|-
 +
|'''Evaluating the Integral:''' &nbsp; Using the earlier steps, we have
 +
|-
 +
|<math>V=\pi \int_{x_1}^{\, x_2} R^2-r^2 \,dx</math>
 +
|-
 +
|&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<math>=\pi \int_{-4}^{\, 5} (25-x^2)^2-(5-x)^2 \,dx</math>
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|-
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|&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<math> =\pi \int_{-4}^{\, 5} 625-50x^2+x^4-(25-10x+x^2) \,dx</math>
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|-
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|&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<math> =\pi \int_{-4}^{\, 5} 600-51x^2+x^4+10x \,dx</math>
 +
|-
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|&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<math> =\pi \int_{-4}^{\, 5} 600-51x^2+x^4+10x \,dx</math>
 +
|-
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|&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<math> =\pi \biggr(600x-51\cdot\frac{x^{3}}{3}+\frac{x^{5}}{5}+10\cdot\frac{x^{2}}{2}\biggr)\biggr|_{x=-4}^{5}</math>
 
|-
 
|-
|We must set the two functions equal, and solve.  More to follow...
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|&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<math> =\frac{15,309}{5} \,\pi.</math>
 
|}
 
|}

Latest revision as of 18:30, 10 March 2015

9BSF1 3a.png

Find the volume of the solid obtained by rotating the area enclosed by and
around the x-axis.

Foundations  
• Choose either shell or washer method.
• Find the appropriate radii.
• Determine the bounds of integration by finding when both functions have the same y value.
• Using the determined values, set up and solve the integral.

Solution:

Step 1:  
Choosing the Approach:   Since we are rotating around the x-axis, the washer method would utilize tall rectangles with dx as their width. This seems like a reasonable choice, as these rectangles would be trapped between our two functions as x varies over the enclosed region, allowing us to solve a single integral. Note that the washer method will require an inner and outer radius, as well as bounds of integration, in order to evaluate the integral
  
Step 2:  
Finding the Radii:  Since our rectangles will be trapped between the two functions, and will be rotated around the x-axis (where y = 0), we find
the inner radius is , represented by the blue line, while
the outer radius is , represented by the red line.
Step 3:  
Finding the Bounds of Integration:   We must set the two functions equal, and solve. If
then by moving all terms to the left hand side and factoring,
so we have -4 and 5 as solutions. These are our bounds of integration.
Step 4:  
Evaluating the Integral:   Using the earlier steps, we have