1. ## Shell Method Help

I've been stuck on these problems and was hoping to have someone solve them so I could see how each one is done. Consider the curves given by y=sinx and cosx
For each of the following problems, you should include a sketch of the region/solid being considered, as well as a labeled representative slice.
1. Set up a definite integral whose value is the exact volume of the solid of revolution generated by revolving about the y-axis
2. Set up a definite integral whose value is the exact volume of the solid of revolution generated by revolving about y=2

Consider the finite region y = 1+ 0.5(x-2)^2 and y = 0.5x^2, and x=0
3. Deterimine an expression involving one or more whose value of the volume of revolution generated by revolving the region R about the y-axis.

2. ## Re: Shell Method Help

Originally Posted by donutstaste
I've been stuck on these problems and was hoping to have someone solve them so I could see how each one is done. Consider the curves given by y=sinx and cosx
For each of the following problems, you should include a sketch of the region/solid being considered, as well as a labeled representative slice.
1. Set up a definite integral whose value is the exact volume of the solid of revolution generated by revolving about the y-axis
2. Set up a definite integral whose value is the exact volume of the solid of revolution generated by revolving about y=2

For #1 and #2, define the x-interval of each region to be rotated ...

Consider the finite region y = 1+ 0.5(x-2)^2 and y = 0.5x^2, and x=0
3. Deterimine an expression involving one or more whose value of the volume of revolution generated by revolving the region R about the y-axis.
$1+0.5(x-2)^2 = 0.5x^2 \implies x = 1.5$

$\displaystyle V = 2\pi \int_0^{1.5} x[1+0.5(x-2)^2 - 0.5x^2] \, dx$