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

1. anonymous

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.

2. anonymous

hey thank u so much