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Expert: - 9/10/2009


Question
Find the volume of the solid generated by revovling the region bounded by the graphs of y=x^2, and  y=20x-x^2 about the line y=102

Answer
To find where the graphs intersect, set x² = 20x - x².
This gives 2x² - 20x = 0, which factors to 2x(x-10)=0.  They cross at x=0 and x=10.

Our first one, y=x², is a concave up parabola with a minimum at (0,0).
The highest the first one gets in this interval is at x=10, whcih is 100, which is less than 102.

The second one is a concave down parabola.
The highest the second one gets, by finding y', is where 20 - 2x = 0, so it is also at x=10.
Notice that it is also at 100 (as it should, since this is where it intersects the first one).

Putting them around the line 102 means we need to find the radius.
For y = x², the radius is 120-y = -x² + 120.
For y = 20x - x², the radius is 120-y = x² - 20x + 102.

The best way to find the volume betwen the two is to find the volume of each and subtract them.

The volume is π∫y²dy.  The two curves are y1 = -x² + 120 and y2 = x² - 20x + 102.
It is known that (y1)² = x^4 - 240x² + 14,400.
It is also known that (y2)² = x^4 - 40x³ + 602x² - 4,080x + 10,400.

The difference (x^4 - 240x² + 14,400) - (x^4 - 40x³ + 602x² - 4,080x + 10,400) =
40x³ - 842x² + 4,080x  + 400.

Integrate that from 0 to 10, then multiply by π.

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