Difference between revisions of "Prototype Calculus Question"
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;" | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | ||
− | ! Foundations | + | ! Foundations |
|- | |- | ||
|• Choose either shell or washer method. | |• Choose either shell or washer method. | ||
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;" | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | ||
− | !Step 1: | + | !Step 1: |
|- | |- | ||
|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. | |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. | ||
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;" | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | ||
− | !Step 2: | + | !Step 2: |
|- | |- | ||
|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 | |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 | ||
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;" | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | ||
− | !Step 3: | + | !Step 3: |
|- | |- | ||
|We must set the two functions equal, and solve. More to follow... | |We must set the two functions equal, and solve. More to follow... | ||
|} | |} |
Revision as of 19:05, 1 March 2015
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: |
---|
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. |
Step 2: |
---|
Since our rectangles will be trapped between the two functions, and will be rotated around the x-axis (where ,) we find |
the inner radius is , represented by the blue line, while |
the outer radius is , represented by the red line. |
Step 3: |
---|
We must set the two functions equal, and solve. More to follow... |