find volume created by rotating area between 2 curves around line

Jun 2009
Calculate the volume found by rotating the area between \(\displaystyle x^3\) and \(\displaystyle \sqrt{x}\) around y=1. I found the intersections of the two functions to be 0 and 1. I am using the slice method. Thinking of how it is outer circle-inner circle I do \(\displaystyle A(x)=\pi(x^3)^2-pi(\sqrt{x})^2\). Since the line of reflection is at one of the intercepts that means I don't have to do the y intercept minus the function, I think. So then I get the formula \(\displaystyle \int_0^1 A(x)\) and that gives me a negative number(Doh)
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MHF Hall of Honor
Feb 2009
It haven't done one of these in over 10 years.
It's important to draw AND note which line you're revolving about.

The formula I believe is \(\displaystyle \pi\int_0^1 (R^2-r^2)dx\).

And by looking at the picture, each radius is 1-y.

So \(\displaystyle R=1-x^3\) is the outer radius and the smaller radius is \(\displaystyle R=1-\sqrt x\).

Hopefully this leads to a postive volume.
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