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 method and got 1250pi. However, I can't figure out how to calculate it using the disk method. The answers should be the same unless I calculated the volume using the shell method incorrectly.
I redid my shells because of a typo, and I got
2π∫[0,5] x(25-((x-2)^3-2)) dx
= 500π
How did you get 1250π? What was your integral?
We need to integrate over x, because the thickness of the shells is dx, not dy.
As for discs,
x = ∛(y+2) + 2
v = π∫[-10,25] πx^2 dy
= π∫[-10,25] (∛(y+2) + 1)^2 dy
= 500π
I also checked my calculation in our previous post
http://www.jiskha.com/display.cgi?id=1385311297
I carelessly dropped the π in my last 3 lines, and should have used my calculator to add up the terms in my last line.
My last line should have been 500π
which then also agrees with Steve's new answer using shells
So you have your two methods,
mine using disks
Steve's using shells
both 500π
To find the volume of the solid using the disk or washer method, you need to follow these steps:
1. Draw a sketch of the region given by the lines and curves. In this case, the region is bounded by the lines x = 0, y = 25, and the curve y = (x - 2)^3 - 2.
2. Determine the limits of integration. Since the region is being revolved about the y-axis, you need to find the y-values where the curves intersect with the line x = 0 and the line x = 25.
To find the lower limit, set x = 0 in the equation y = (x - 2)^3 - 2:
y = (-2)^3 - 2 = -6
To find the upper limit, set x = 25 in the equation y = (x - 2)^3 - 2:
y = (25 - 2)^3 - 2 = 15023
Therefore, the limits of integration are y = -6 to y = 15023.
3. Determine the radius of the disc or washer at each value of y. Since we are revolving the region about the y-axis, the radius is given by the x-coordinate of the curve y = (x - 2)^3 - 2.
Rearranging the equation, we get:
x = (y + 2)^(1/3) + 2
4. Determine the volume of each disc or washer. The volume of a disc or washer is given by the formula: V = πr^2Δy, where r is the radius and Δy is the change in y.
The change in y is determined by the limits of integration, so Δy = 15023 - (-6) = 15029.
5. Integrate the volume expression. The volume of the solid can be found by integrating the volume of each disc or washer over the interval of integration.
V = ∫[from -6 to 15023] π((y + 2)^(1/3) + 2)^2 dy
6. Evaluate the integral to find the volume of the solid. Use techniques such as substitution or integration by parts to evaluate the integral.
After integrating, you should get the volume of the solid.
In your case, you calculated the volume using the shell method, which would yield a different result than the disk or washer method. Therefore, it's important to check your calculations or perform the integration using the disk or washer method as described above to compare the results.