1. Draw your curve, and then shade in the area you want to find.Similar to before, but now you're shading the right half of the ellipse.
2. Figure out what kind of a shape you'll get.Just gotta look up again. I believe in you! ;) Remember, this will be the hardest step - suddenly having to visualise in the third dimension is not easy when you've spent all this time in the second.
3. Figure out what your limits a and b are.Limits are -b to +b. If you had trouble with this, go backwards and learn how to find the area between a curve and the y-axis.
4. Figure out what axis you want to rotate around.Going around y, giving us a dy. So, our formula is
5. Have I got everything?Close - but once again, we need to isolate either x or y. As you can see just above this, we want x^2, so we get
Now, we plug and play:
Which is, once again, what we predicted!
Also, for fun - if we assume these two volumes are equal, we imply that
, and we get the volume for a sphere. How cool is that?