Question

How do you find the volume of the solid generated by revolving the region bounded by the curves y=x^3 and y=x^4 rotated about the y-axis?

Answer 1

This is something that can be done two ways---the earlier way and the Shell Method. The Shell Method isn't too difficult to apply.

In this method, since you are rotating about the y-axis, your thickness `f(x)` is the function farther right minus the function farther left. If you get a negative volume at the end, you know you did it backwards.

Your graph basically looks like:

graph{(x^3 - y)(x^4 - y)sqrt(0.5^2 - (x-0.5)^2)/sqrt(0.5^2 - (x-0.5)^2) <= 0.00 [-1, 2, -0.095, 2]}

from `x = 0` to `x = 1`.

The formula uses the idea of circumference (the `2pix`) with the varying thickness `f(x)` of the solid along the vertical direction to build the solid by stacking shells vertically, while the radius `x` indicates the distance from the axis of rotation; since the axis of rotation is the y-axis, the radius is simply `x`, spanning `0` to `pm1`.

`V = int2pixf(x)dx`

`= 2piint x(x^3 - x^4)dx`

`= 2piint x^4 - x^5dx`

`= 2pi [1/5x^5 - 1/6x^6]|_(0)^(1)`

`= 2pi [(1/5 - 1/6) - 0]`

`= color(blue)(pi/15 "u"^3)`