- #1
jld592
- 4
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So I was doing my calc homework when I stumbled upon this thought:
lets say you were trying to find the solid of rotation of y=x^2 around the x-axis over the interval [-2,2].
the traditional method would entail pi * integral from -2 to 2 of (x^2)^2 dx
while this is easier for a simple graph, squaring the integrand would make a complex integral much more complex to do by hand.
Wouldn't it be simpler to find the average value of the function over the interval and rotate the line segment y = average value over the interval around the x-axis creating a cylinder.
So for the example problem this would be the cylinder created by rotating the line segment y = 4/3 where -2<x<2 around the x axis.
However I tried this and found that the traditional method yielded 64pi / 5 while my method yielded 64pi / 9. I think this has something to do with the fact that you would have to average the areas of the circles, but my brain still tells me that my method should work
lets say you were trying to find the solid of rotation of y=x^2 around the x-axis over the interval [-2,2].
the traditional method would entail pi * integral from -2 to 2 of (x^2)^2 dx
while this is easier for a simple graph, squaring the integrand would make a complex integral much more complex to do by hand.
Wouldn't it be simpler to find the average value of the function over the interval and rotate the line segment y = average value over the interval around the x-axis creating a cylinder.
So for the example problem this would be the cylinder created by rotating the line segment y = 4/3 where -2<x<2 around the x axis.
However I tried this and found that the traditional method yielded 64pi / 5 while my method yielded 64pi / 9. I think this has something to do with the fact that you would have to average the areas of the circles, but my brain still tells me that my method should work
