Volume of a sphere formula?

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Volume of a sphere formula?

The main idea is that the volume of a sphere comes from stacking many thin disks, so the volume grows with the cube of the radius. At a height z from the center, the cross-sectional disk has radius sqrt(r^2 − z^2), so its area is π(r^2 − z^2). Integrate these disks from the bottom to the top (z goes from −r to r): V = ∫_{−r}^{r} π(r^2 − z^2) dz = π [r^2 z − z^3/3]_{−r}^{r} = π[(r^3 − r^3/3) − (−r^3 + r^3/3)] = π(4r^3/3) = 4/3 π r^3. So the volume is 4/3 π r^3. This also matches the idea that doubling the radius multiplies volume by 8.

The main idea is that the volume of a sphere comes from stacking many thin disks, so the volume grows with the cube of the radius. At a height z from the center, the cross-sectional disk has radius sqrt(r^2 − z^2), so its area is π(r^2 − z^2). Integrate these disks from the bottom to the top (z goes from −r to r):

V = ∫{−r}^{r} π(r^2 − z^2) dz = π [r^2 z − z^3/3]{−r}^{r} = π[(r^3 − r^3/3) − (−r^3 + r^3/3)] = π(4r^3/3) = 4/3 π r^3.

So the volume is 4/3 π r^3. This also matches the idea that doubling the radius multiplies volume by 8.

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