Question

Use the shell method to find the volume of the solid generated by revolving the region bounded by y = 9x-8, y=sqrt(x), and x = 0 about the y-axis in cubic units.

Answer 1

I have the problem set up as this: \[2\Pi \int\limits_{0}^{1}(9x-8)(\sqrt{x})dx\]

Answer 2

When I go to take the antiderivatives, I come up with \[(18/5)x ^{(5/2)}-(16/3)x^(3/2) from 0 \to 1\]

Answer 3

Which might be where I went wrong. Can someone take a look at my integration and see if I made a mistake somewhere? The answer that I came up with with the above equation was 52pi/15, however the actual answer should be 14pi/5

Answer 4

That's the shape right? Notice that when you make the shells of cylinders that you actually have one integration from 0 to the point of intersection of the functions minus the integration from where the linear function intersects the x-axis to the point it intersects the square root of x.