# Volumes by Cylindrical Shells

Hi, I was hoping someone could check my work on a few problems and get me started on a few others. It involves definite integration, so I'm gonna use (a,b)S as an integration symbol and P for pi.

These are the ones I need checked:
1. Use cylindrical shells to find the volume of the solid generated when the region enclosed by the given curves is revolved about the x-axis.
x=2y, y=2, y=3, x=0
2P*(2,3)Sy(2y)dy = 4P*(2,3)S(y^2)dy = 4P/3*[y^3](2,3) = 76P/3

2. Use cylindrical shells to find the volume of the solid that is generated when the region that is enclosed by y=1/x^3, x=1, x=2, y=0 is revolved about the line x=-1
2P*(1,2)S(x/(x-1)^3) = -2P*[(1-2x)/(2(x-1)^2)](1,2) =
I'm stuck here, because putting 1 into the equation puts a zero in the denominator.

3. (a) Find the volume V of the solid generated when the region bounded by y=1/(1+x^4), y=0, x=1, and x=b (b>1) is revolved about the y-axis.
(b) Find lim(b->+infinity) V
(a) 2P*(1,b)S(x/(1+x^4)) = 2P*[(x^2)/2 - 1/(2x^2)](1,b) = 2P(.5b^2 - 1/(2b^2))
(b) Infinity

4. The base of a certain solid is the region enclosed by y = x^.5, y=0, and x=4. Every cross section pependicular to the x-axis is a semicircle with its diameter across the base. Find the volume of the solid.
P/16*(0,4)Sxdx = .5P

These are the ones where I don't even know where to start:
5. The region enclosed between the curve y^2=kx and the line x=.25k is revolved about the line x=.5k. Use cylindrical shells to find the volume of the resulting solid. (Assume k>0)

6. Use cylindrical shells to find the volume of the torus obtained by revolving the circle x^2 + y^2 = a^2 about the line x=b, where b>a>0. [Hint: It may help in the integration to think of an integral as an area.]

Much thanks to anyone who can give me any help. I really appreciate it.

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Math Is Hard
Staff Emeritus
The first one looks good, but that's all I have had time to look at. I am an old lady and very slow! And it's been a while since i've done any shelling!