Use shells to find the volume of a sphere of radius r

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MarkSA

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Hello,

Use cylindrical shells to find the volume of the solid.
1) A sphere of radius r.

I did a problem similar to this (a cone) and didn't have too much trouble, but i'm kind of stumped on this one.

I used the formula for and drew a semi circle.. I figured I would need that to do this one. y = sqrt(r^2 - x^2) and then rotated the semi circle around the x-axis to get sphere... I then drew the shell, with the approximating piece laying horizontally like so:
|--------|

So this gets me, V = 2pi * integral from 0 to r of: [y * sqrt(r^2 - y^2)]dy
with y being the radius and sqrt(r^2 - y^2) being the height of the shell.

This is where I run into trouble. How can I get the antiderivative of y*[sqrt(r^2-y^2)] to solve this problem? I can't see how to do it easily, which leads me to believe I may have set this problem up wrong... since none of the others in the homework caused this trouble.

Thanks.
 
How can I get the antiderivative of y*[sqrt(r^2-y^2)] to solve this problem?
Click to expand...

how about using the substitution u = r[sup:2ga59hec]2[/sup:2ga59hec] - y[sup:2ga59hec]2[/sup:2ga59hec] ?

remember, r[sup:2ga59hec]2[/sup:2ga59hec] is just a constant.
 
Hello,

thanks for the reply. I tried the u-substitution method but i'm oddly ending up with the wrong answer when I evaluate the problem.

My final answer ends up being 2/3 * pi * r^3
It would seem i'm missing a multiplication by 2 somewhere. I think i'm evaluating the integral properly though...

Here is the formula for semi circle: x = sqrt(r^2 - y^2), it is then rotated around the x axis.

Here is the integral I start with:

V = integral from 0 to r of: 2*pi*y*(sqrt(r^2 - y^2))dy
Let u = r^2 - y^2
du = -2ydy
-1/2*du = ydy
-pi * integral from 0 to r of: u^(1/2)du
= 2/3 * pi * r^3

Any ideas what the problem could be?
 
Sorry but could you explain where in the process I would multiply by 2.

I took a semicircle equation and rotated it around the x-axis. Shouldn't the standard way of getting the volume with shells work? with the formula:
integral from a to b of: 2*pi*x * f(x) * dx

How do you know when you need to multiply it by 2?

Thanks
 
the limits of integration from 0 to r result in the volume of a hemisphere.

to get the volume of the entire sphere, your limits of integration should be -r to r or, using symmetry, you can use 0 to r and double the resulting volume.
 
You can use washers:

\(\displaystyle {\pi}\int_{-r}^{r}(r^{2}-x^{2})dx\)

That's probably why they wanted you to use shells. It's a we bit harder than the washers method.
 
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