Question

find volume of a sphere of radius r by slicing

Answer 1

Nath, to do this, consider a sphere centered at (0,0). Take a thin slice somewhere by cutting parallel to the y-axis, 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 dead-on at the sphere, you'll see that that y-value 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\[y^2=r^2-x^2\]and so your formula for volume of the slice becomes,\[\delta V \approx \pi (r^2-x^2) \delta x\]In the limit, as delta x approaches 0, we get an infinitesimally thin slice, that we can add up using integration:\[V=\pi \int\limits_{-r}^{r}r^2-x^2 dx=\pi [r^2x-\frac{x^3}{3}|_{-r}^{r}\]\[=\pi (r^3-\frac{r^3}{3}-(-r^3+\frac{r^3}{3}))=\frac{4 \pi }{3}r^3\]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.

Answer 2

hey thank u so much