Chapter 7, Problem 21RE

i.

To determine

To calculate: To solve a volume of the solid generated by revolving the region

Answer to Problem 21RE

  $V=\frac{32\pi }{3}$ .

Explanation of Solution

Given information: The given information as follows:

a volume of the solid generated by revolving the region enclosed by the parabola $y^2=4x$ and the line $y=x$ about $thex-axis$ .

Calculation:

The volume of the solid generated by revolving the region bounded by the given parabola and the straight line can be calculated as:

  $V=V_2\left(R\right)-V_1\left(R\right)$

Here, $y=x_1,\frac{y^2}{4}=x_2$ .

  $\begin{array}{l}{V}_1\left(R\right)=2\pi {\displaystyle {\int }_0^R{f}_1\left(R\right)}rdr\\ V_2\left(R\right)=2\pi {\displaystyle {\int }_0^R{f}_2\left(R\right)}rdr\end{array}$

  $R$ is such that $f_1\left(R\right)=f_2\left(R\right)$ giving $R=4$ .

The volume can be calculated as follows:

  $\begin{array}{c}V=2\pi {\displaystyle {\int }_0^4\left(\frac{r^2}{4}-r\right)}dy\\ V=2\pi {\left[\frac{r^3}{12}-\frac{r^2}{2}\right]}_0^4\\ V=2\pi \left[\frac{64}{12}-\frac{16}{2}\right]\\ V=\frac{32\pi }{3}\end{array}$

ii.

To determine

To calculate: To solve a volume of the solid generated by revolving the region

Answer to Problem 21RE

  $V=\frac{128\pi }{15}$ .

Explanation of Solution

Given information: The given information as follows:

a volume of the solid generated by revolving the region enclosed by the parabola $y^2=4x$ and the line $y=x$ about $they-axis$ .

The volume of the solid generated by revolving the region bounded by the given parabola and the straight line can be calculated as:

  $V={\displaystyle {\int }_a^b\pi \left(x_1{}^2-x_2{}^2\right)}dy$

Here, $y=x_1,\frac{y^2}{4}=x_2$ .

The volume can be calculated as follows:

  $\begin{array}{c}V=\pi {\displaystyle {\int }_0^4\left(y^2-\frac{y^4}{16}\right)}dy\\ V=\pi {\left[\frac{y^3}{3}-\frac{y^5}{80}\right]}_0^4\\ V=\pi \left[\frac{64}{3}-\frac{1024}{80}\right]\\ V=64\pi \left[\frac{1}{3}-\frac{1}{5}\right]\\ V=\frac{128\pi }{15}\end{array}$

iii.

To determine

To calculate: To solve a volume of the solid generated by revolving the region

Answer to Problem 21RE

  $V=\frac{140\pi }{3}$ .

Explanation of Solution

Given information: The given information as follows:

a volume of the solid generated by revolving the region enclosed by the parabola $y^2=4x$ and the line $y=x$ about $thelinex=4$ .

The volume of the solid generated by revolving the region bounded by the given parabola and the straight line can be calculated as:

  $V=V_2\left(R\right)-V_1\left(R\right)$

Here, $4-y=r_1,4-\frac{1}{2}\sqrt{y}=r_2$ .

  $\begin{array}{l}{V}_1\left(R\right)=2\pi {{\displaystyle {\int }_0^R\left[f_1\left(R\right)\right]}}^{2}dy\\ V_2\left(R\right)=2\pi {{\displaystyle {\int }_0^R\left[f_2\left(R\right)\right]}}^{2}dy\end{array}$

  $\begin{array}{l}V=2\pi {{\displaystyle {\int }_0^4\left(4-\frac{1}{2}\sqrt{y}\right)}}^2-{\left(4-y\right)}^{2}dy\\ V=2\pi {\displaystyle {\int }_0^4\left(-y^2+\frac{33}{4}y-4\sqrt{y}\right)}dy\\ V=\frac{140\pi }{3}\end{array}$