# Math (Calculus)

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the y-axis.

y=4-x^2, y=0

1. 👍 0
2. 👎 0
3. 👁 72
asked by Sherman
1. The volume of a shell of radius r, height h and thickness dy (because revolving around the y-axis) is

2πrh dx

Now, our plane region is symmetric about the y-axis, so revolving all of it it is the same as revolving only the part for x>=0. o, we'll take the volume to be

∫[0,2] 2πrh dx
where r=x and h=y
= 2π∫[0,2] x(4-x^2) dx
= 2π(2x^2 - 1/4 x^4) [0,2]
= 8π

You can check using shells of thickness dy. The volume is then

∫[0,4] πr^2 dy
where r=x
= π∫[0,4] (4-y) dy
= π(4y - 1/2 y^2) [0,4]
= 8π

1. 👍 0
2. 👎 0
posted by Steve

## Similar Questions

1. ### Calculus

Use the shell method to set up, but do not evaluate, an integral representing the volume of the solid generated by revolving the region bounded by the graphs of y=x^2 and y=4x-x^2 about the line x=6. I had the shell radius as

asked by Ryan on March 7, 2011
2. ### calculus

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the y-axis. y = 6x y = 18 x = 0

asked by MAM on January 24, 2012
3. ### calculus

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the x-axis: Y=sqrt(x+2), y=x,y=4

asked by carlton on May 2, 2012
4. ### Math

Let A denote the portion of the curve y = sqrt(x) that is between the lines x = 1 and x = 4. 1) Set up, don't evaluate, 2 integrals, one in the variable x and one in the variable y, for the length of A. My Work: for x:

asked by Daisy on January 25, 2017
5. ### calculus

using the method of shells, set up, but don't evaluate the integral, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. Y=e^x, x=0, y=2, about y=1

asked by marks on March 16, 2014
6. ### Calculus

Write the integral in one variable to find the volume of the solid obtained by rotating the first-quadrant region bounded by y = 0.5x^2 and y = x about the line x = 7. I have to use the volume by disks method, but I'm confused

asked by Anonymous on May 11, 2018
7. ### Calculus

Use the Shell Method to compute the volume V of the solid obtained by rotating the region enclosed by the graphs of the functions y = x^2, y = 8 − x^2,and x = 1/2 about the y-axis. Here is how I set up the integral: 2 pi

asked by Anonymous on October 31, 2015
8. ### Calculus - Volume By Integration

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the y-axis: y=(x-2)^3-2, x=0, y=25 Solve by either the disk or washer method. I calculated the volume using the shell

asked by Nicole on November 25, 2013
9. ### calculus

8). Part 1 of 2: In the solid the base is a circle x^2+y^2=16 and the cross-section perpendicular to the y-axis is a square. Set up a definite integral expressing the volume of the solid. Answer choices: integral from -4 to 4 of

asked by Sally on April 14, 2013
10. ### Math

Hi, I have to do a project on the Shell Method in my Geometry class. I have been trying to find information on that topic, but have no luck... can someone point me to the right direction as of what keyword to search for, or is

asked by Billy on April 14, 2007

More Similar Questions