Prototype Calculus Question

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9BSF1 3a.png

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

• 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.


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, allowing us to solve a single 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 ), 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
so we have roots -4 and 5. These are our bounds of integration.
Step 4:  
Evaluating the Integral:   Using the earlier steps, we have