Math 152: Worksheet 5

Solids of Revolution - Shell Method

Learning Problems

These problems should be completed on your own. If you need hints on solving a problem, there are some provided with each problem. Click on the word "hint" to view it and again to hide it. They go in increasing order of helpfulness, with the last hint mostly giving away how to do the problem. Try to work from the earlier hints to the later ones, as this will give you the practice you need to succeed in this class.

Problem 1

Find the volume of the solid of revolution obtained by revolving the graph of $y = 2x^2$ around the $y$-axis over the range $[1,4]$.

• Hint 1

  Draw a picture of the region. What should the radius and height be for this region?

• Hint 2

  From that, set up the integral using the formula for the shell method. (This should be an integral in $x$).

• Hint 3

  The integral to find the volume is $\displaystyle 2\pi \int_1^4 x(2x^2) \ dx$

• Answer

  $\displaystyle 255\pi$

Problem 2

Find the volume of the solid of revolution obtained by revolving the region enclosed between the graphs of $y = 3 - x$ and $y = x^2 - 3$ around the line $x = 5$.

• Hint 1

  Sketch out a picture. Draw in the segment that is being rotated and try to set up the integral.

• Hint 2

  What should the height and radius be for doing this via the Shell Method?

• Hint 3

  The radius should be $5-x$, where the height is the difference between the two functions.

• Hint 4

  The integral is $\displaystyle 2\pi \int_{-3}^2 (5-x)(3-x-(x^2 - 3))\ dx$

• Answer

  $\displaystyle \frac{1375}{6}\pi$

Problem 3

Find the volume of the solid of revolution obtained by revolving the region in the first quadrant between the graph of $y = x^2 + 4$ and the line $y = 13$ around the $x$ axis using the Shell Method.

• Hint 1

  If we're doing this around the $x$ axis via the Shell Method, this means we need an integral in $y$. Draw out the picture and the segment that is being rotated.

• Hint 2

  What is the radius and height for this setup? What are the bounds on the integral?

• Hint 3

  The integral should be $$ 2\pi \int_4^{13} y(\sqrt{4-y})\ dy $$

• Answer

  $\displaystyle \frac{1692}{5} \pi $

Problem 4

Find the volume of the solid of revolution obtained by revolving the region in the first quadrant between the graph of $y = x^2 + 4$ and the line $y = 13$ around the $y$ axis using the Shell Method.

• Hint 1

  Since we want to revolve around the $y$ axis with shells, this requires an integral in $x$. Draw out the picture and the segment being rotated.

• Hint 2

  What are the radius and height for this rotation? What are the bounds on the integral?

• Hint 3

  The integral should be $$ 2\pi \int_0^{3} x(13 - (x^2 + 4))\ dx $$

• Answer

  $\displaystyle \frac{81}{2} \pi $

Problem 5

Consider the solid of revolution formed by revolving the region between the curves $y=x^2$, $y = 4$ and $x=1$ around the line $x = -2$. Work out the answer using both the washer method and the shell method.

• Hint 1

  Sketch out the region and look at what the segments are that will be rotated for each method.

• Hint 2

  Washer Method: The inner radius should be the straight line, and outer radius the curve.

• Hint 3

  Shell Method: The radius should be $x+2$, and the height should be the gap between the curve and the horizontal line.

• Answer

  $\displaystyle \frac{67}{6} \pi$

Submission Problems

Problem 1

Find the volume of the solid of revolution obtained by revolving the region between the graphs of $x = 9 - y^2$ and $x = 5$ around the line $y = 8$.

Problem 2

Find the volume of the solid of revolution obtained by revolving the region between the graphs of $y = x^2 + 1$, $y = 1-x$, and the lines $x=1$ and $x=5$ around the line $x = -2$.