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zairaefh6496

13.03.2022

The volume of a sphere is given by:

$V= \frac{4}{3} \pi r^{3}$

So, we need to deduct this equation. We will walk through Calculus on the concept of a solid of revolution that is a solid figure that is obtained by rotating a plane curve around some straight line (the axis of revolution) that lies on the same plane. We know from calculus that:

$V=\pi \int_{a}^{b}[f(x)]^{2}dx$

Then, according to the concept of solid of revolution we are going to rotate a circumference shown in the figure, then:

$x^{2}+y^{2}=r^{2}$

Isolationg y:

$y= \sqrt{r^{2}-x^{2}}$

So,

$f(x)=y=\sqrt{r^{2}-x^{2}}$

$V=\pi \int_{a}^{b}[\sqrt{r^{2}-x^{2}}]^{2}dx$

$V=\pi \int_{a}^{b}(r^{2}-x^{2})dx$

being -r and r the limits of this integral.

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

Solving:

$V=\pi[r^{2}x-\frac{x^{3}}{3}]\right|_{-r}^{r}$

Finally:

$V=\pi(r^{3}-\frac{r^{3}}{3})-\pi(-r^{3}+\frac{r^{3}}{3})= \frac{4}{3} \pi r^{3}$

Drag and drop an answer to each box to correctly complete the explanation for deriving the formula f

Drag and drop an answer to each box to correctly complete the explanation for deriving the formula f

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Drag and drop an answer to each box to correctly complete the explanation for deriving the formula for the volume of a sphere.

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