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Chapter 6: Q. 64 (page 512)

Use a definite integral to prove that a sphere of radius r has volume given by the formula $V = \frac{4}{3} {蟺谤}^{3}$ .

Short Answer

Expert verified

It has been proved that sphere has volume $\frac{4}{3} {蟺谤}^{3}$

Step by step solution

01

Step 1. Given information

The radius of the sphere is $r$

The volume to be proved of the sphere is $V = \frac{4}{3} {蟺谤}^{3}$

02

Step 2. Proof of the volume of the sphere.

Draw a diagram of the sphere:

Here a circular slice of thickness $d x$ has been cut at a distance of $x$ from the center of the sphere.

The radius of the slice is:

$A B = \sqrt{r^{2} - x^{2}}$

The area of the slice is $蟺 {(AB)}^{2} = 蟺 (r^{2} - x^{2})$

The volume of the slice is $d V = 蟺 (r^{2} - x^{2}) dx$

Hence the volume of the sphere is :

$V = {鈭�}_{- r}^{r} 蟺 (r^{2} - x^{2}) dx = 蟺 {鈭�}_{- r}^{r} (r^{2} - x^{2}) dx = 2 蟺 {鈭�}_{0}^{r} (r^{2} - x^{2}) dx = 2 蟺 {[r^{2} x - \frac{x^{3}}{3}]}_{0}^{r} = 2 蟺 (r^{3} - \frac{r^{3}}{3} - 0) = 2 蟺 (\frac{2 r^{3}}{3}) = \frac{4}{3} {蟺谤}^{3}$

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Most popular questions from this chapter

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29鈥�52.

35. $\frac{d y}{d x} = \frac{3}{y}, y (0) = 2$

Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants.

$\frac{d y}{d x} = x y^{2}$

In the process of solving $\frac{d y}{d x} = y$ by separation of variables, we obtain the equation $| y | = e^{x + C}$ . After solving for $y$ , this equation becomes $y = A e^{x}$ . How is $A$ related to $C$ ? What happened to the absolute value?

Each of the definite integrals in Exercises 19鈥�24 represents the volume of a solid of revolution obtained by rotating a region around either the x- or y-axis. Find this region.

$蟺 {鈭�}_{1}^{5} {(\frac{y - 1}{2})}^{2} dy$

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29鈥�52.

32. $\frac{d y}{d x} = - 5 y, y (0) = 2$

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