Say I have a sphere and it has a radius of 3. How could I find it's volume using the disk method and integration?
By disk method I assume you mean using cylindrical coordinates. In this case you need to think about your limits on z. At what z is the top and bottom of the circle? At .
So what is the equation of our circle? Its:
We know from polar coordinates that . Then the equation of our circle is now:
But in polar coordinates, a negative r doesn't make sense so we will say:
So now what is the area of any disk in our circle? It is:
But we want to add up all our disks, so we multiply them by a little bit of height, dz, and then integrate over z also:
Does it work? Lets evaluate it, we know we should get
Indeed, if you went through and solved the triple integral you would get .
Since you are only using 1/2 of a circle, use the equation:
Can you take it from here?