How to find the volume of a sphere using integration?

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How to find the volume of a sphere using integration?

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Answer 1 · 2014-09-18T20:10:35

Since a sphere with radius $r$ $r$ can be obtained by rotating the region bounded by the semicircle $y = \sqrt{r^{2} - x^{2}}$ $y=sqrt{r^2-x^2}$ and the x-axis about the x-axis, the volume $V$ $V$ of the solid can be found by Disk Method.

$V = π \int_{- r}^{r} {(\sqrt{r^{2} - x^{2}})}^{2} d x$ $V=pi int_{-r}^r(sqrt{r^2-x^2})^2dx$

by the symmetry about the y-axis,

$= 2 π \int_{0}^{r} (r^{2} - x^{2}) d x$ $=2piint_0^r(r^2-x^2)dx$

$= 2 π {[r^{2} x - \frac{x^{3}}{3}]}_{0}^{2}$ $=2pi[r^2x-x^3/3]_0^r$

$= 2 π (r^{3} - \frac{r^{3}}{3})$ $=2pi(r^3-r^3/3)$

$= \frac{4}{3} π r^{3}$ $=4/3pir^3$

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How to find the volume of a sphere using integration?

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