Chapter 2.3, Problem 50E

To determine

The reason behind the rate of change of volume with respect to radius is area.

Explanation of Solution

Given information:

The radius of the sphere is $r$ , volume of sphere is $\frac{4}{3}\pi r^3$ and surface area is $4\pi r^2$ .

Concept used:

The formula used is $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$

Calculation:

The sphere has been divided into small strips.

The area of each strip will be equal to the area of the sphere.

That is $A=4\pi r^2$

The width of each strip is change in radius.

That is $w=dr$

The volume of a sphere is given by the formula $V=\frac{4}{3}\pi r^3$

The change in volume be $dV$ .

So the change in volume is $4\pi r^2.dr$

Equate both the equations $dV=4\pi r^2.dr$

  $\frac{dV}{dr}=4\pi r^2$

Conclusion: Therefore, the change in volume with respect to radius is the area.