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anonymous
 5 years ago
find volume of a sphere of radius r by slicing
anonymous
 5 years ago
find volume of a sphere of radius r by slicing

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anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Nath, to do this, consider a sphere centered at (0,0). Take a thin slice somewhere by cutting parallel to the yaxis, and let that slice have a small thickness, \[\delta x\]Then the approximate volume of that slice is given by\[\delta V \approx \pi y^2 \delta x\]where y is the approximate radius of the circle of the slice you've picked. If you look deadon at the sphere, you'll see that that yvalue is that value contained in the formula for the circle that surrounds the sphere, namely,\[x^2+y^2=r^2\]This means then that\[y^2=r^2x^2\]and so your formula for volume of the slice becomes,\[\delta V \approx \pi (r^2x^2) \delta x\]In the limit, as delta x approaches 0, we get an infinitesimally thin slice, that we can add up using integration:\[V=\pi \int\limits_{r}^{r}r^2x^2 dx=\pi [r^2x\frac{x^3}{3}_{r}^{r}\]\[=\pi (r^3\frac{r^3}{3}(r^3+\frac{r^3}{3}))=\frac{4 \pi }{3}r^3\]Here, the limits of integration were taken from r to r, since our slices start from x=r and end at x=r. Hope this helped. Ask if you need anything more.
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