Chapter 6.2, Problem 47E

a.

To determine

Find the volume enclosed by the cylinder and prove $V_{s_1}=V_{s_2}$

Answer to Problem 47E

The volume enclosed by the cylinder and proved $V_{s_1}=V_{s_2}$

Explanation of Solution

Calculation:

Let $A\left(S_1\right)\text{ and }A\left(S_2\right)$ represent the areas of the cross- sections of the solids $S_1\text{ and }{S}_2$

By the definition of volume by cylindrical shells, know that

  $\begin{array}{l}A\left(S_1\right)\text{ and }A\left(S_2\right)\\ V_{s_1}={\displaystyle \underset{a_1}{\overset{b_1}{\int }}A\left(S_1\right)}dx\\ V_{s_2}={\displaystyle \underset{a_2}{\overset{b_2}{\int }}A\left(S_2\right)}dx\\ \text{assume that }A\left(S_1\right)=A\left(S_2\right)\end{array}$

Because the planes of the cross-sections are parallel.

To know that bounds of the integrals are equal (i.e. $a_1=a_2\text{ and }{b}_1=b_2$ )

Since $A\left(S_1\right)=A\left(S_2\right)$ , $a_1=a_2\text{ and }{b}_1=b_2$ then

  $\begin{array}{l}{\displaystyle \underset{a_1}{\overset{b_1}{\int }}A\left(S_1\right)}dx={\displaystyle \underset{a_2}{\overset{b_2}{\int }}A\left(S_2\right)}dx\\ \therefore V_{s_1}=V_{s_2}\end{array}$

Thus, the volume enclosed by the given integral $V_{s_1}=V_{s_2}$

b.

To determine

Find the volume enclosed by the given cylinder.

Answer to Problem 47E

The volume enclosed by the given cylinder is $V=\pi r^{2}h$.

Explanation of Solution

Use Cavalier’s principle to say that the volume of the cylinder that‘s in the figure is the same as that of a right circular cylinder with radius $r$ and height $h$ . So $V=\pi r^{2}h$

Thus, the volume enclosed by the given cylinder $V=\pi r^{2}h$.