Finding the volume of a solid revolution (shell method)

In summary, the conversation discusses finding the volume of a solid revolution S formed by rotating the area enclosed by the function f(x)=9-x^2 and y>=0 around the vertical line x=7. The speaker used the shell method and found the indefinite integral to be 2∏∫(7-x)(9-x^2)dx = 2∏∫(63-9x-7x^2+x^3)Δx, but was unsure of the interval for the integration. They also tried using the disc method and got the answer 801∏/2. Another person suggests thinking about the description of the area A and using a graph to determine the limits for the integral, which should
  • #1
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Let f(x)=9-x^2. Let A be the area enclosed by the graph y=f(x) and the region y>=0.
Suppose A is rotated around the vertical line x=7 to form a solid revolution S.
So, using the shell method, I was able to find the indefinite integral used.
I found the shell radius to be (7-x) and the shell height to be (9-x^2)
Therefore, the volume is 2∏∫(7-x)(9-x^2)dx = 2∏∫(63-9x-7x^2+x^3)Δx
However, I am just unsure for what the interval should be. For whether it should be [-3,3] or [0,7]. So, which one is it?
I also tried using the disc method, with π ∫ [7² - (9 - y)] dy from 0 to 9 and got the answer 801∏/2. Is that right?
Any help is appreciated. Thanks.
 
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  • #2
To find your limits, think about the description of the area A. It is the area enclosed by the function and the region y >= 0. Picture this area on a graph (make a sketch if it helps), and it should be pretty clear that the integral should go from x=-3 to x=3. Using the other method, your formulas for your radii aren't quite right. Your outer radius should be [tex]7+\sqrt{9-y}[/tex] and your inner radius should be [tex]7-\sqrt{9-y}[/tex]. Again, picture the situation on a graph (for this one, it might be especially helpful to actually draw a quick sketch) and you should see why you get that.
 

What is the shell method for finding the volume of a solid revolution?

The shell method is a mathematical technique used to find the volume of a three-dimensional solid that is formed by rotating a two-dimensional shape around an axis. It involves finding the volume of thin cylindrical shells and then adding them together to get the total volume of the solid.

What are the steps involved in using the shell method to find the volume of a solid revolution?

The first step is to identify the axis of rotation and the boundaries of the shape being rotated. Then, a representative shell is selected, and the radius and height of the shell are determined. Next, the volume of the shell is calculated using the formula V = 2πrhΔx. Finally, the volume of all the shells is added together to get the total volume of the solid.

What types of shapes can be used with the shell method to find volume?

The shell method can be used for any shape that can be rotated around an axis, such as circles, rectangles, and triangles. However, it is most commonly used for shapes that have a simple equation, such as y = f(x) or x = g(y).

What is the difference between using the shell method and the disk method to find the volume of a solid revolution?

The disk method involves slicing the solid into thin disks and finding the volume of each disk, then adding them together. The shell method, on the other hand, involves finding the volume of thin cylindrical shells and adding them together. The shell method is generally preferred for more complex shapes, while the disk method is more suitable for simpler shapes.

Can the shell method be used to find the volume of a solid revolution with a hole or missing section?

Yes, the shell method can still be used for solids with holes or missing sections. In this case, the volume of the hole or missing section must be subtracted from the total volume calculated using the shell method formula. This can be done by finding the volume of the hole or missing section using the appropriate method (such as the disk method) and subtracting it from the total volume.

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