find volume created by rotating area between 2 curves around line

Jun 2009
251
3
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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matheagle

MHF Hall of Honor
Feb 2009
2,763
1,146
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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