Chapter 7.3, Problem 56E

(a)

To determine

The correct option for the integral that describes the volume of a right circular cone along with the dimensions of the torus. The options are:

The integrals:

(i) $2\pi {\displaystyle \underset{0}{\overset{r}{\int }}hxdx}$

(ii) $2\pi {\displaystyle \underset{0}{\overset{r}{\int }}hx\left(1-\frac{x}{r}\right)dx}$

(iii) $2\pi {\displaystyle \underset{0}{\overset{r}{\int }}2x\sqrt{r^2-x^2}dx}$

(iv) $2\pi {\displaystyle \underset{0}{\overset{b}{\int }}2ax\sqrt{1-\frac{x^2}{b^2}}dx}$

(v) $2\pi {\displaystyle \underset{-r}{\overset{r}{\int }}2\left(R-x\right)\sqrt{r^2-x^2}dx}$

(b)

To determine

The correct option for the integral that describes the volume of a right circular cone along with the dimensions of the torus. The options are:

The integrals:

(i) $2\pi {\displaystyle \underset{0}{\overset{r}{\int }}hxdx}$

(ii) $2\pi {\displaystyle \underset{0}{\overset{r}{\int }}hx\left(1-\frac{x}{r}\right)dx}$

(iii) $2\pi {\displaystyle \underset{0}{\overset{r}{\int }}2x\sqrt{r^2-x^2}dx}$

(iv) $2\pi {\displaystyle \underset{0}{\overset{b}{\int }}2ax\sqrt{1-\frac{x^2}{b^2}}dx}$

(v) $2\pi {\displaystyle \underset{-r}{\overset{r}{\int }}2\left(R-x\right)\sqrt{r^2-x^2}dx}$

(c)

To determine

The correct option for the integral that describes the volume of a right circular cone along with the dimensions of the torus. The options are:

(i) $2\pi {\displaystyle \underset{0}{\overset{r}{\int }}hxdx}$

(ii) $2\pi {\displaystyle \underset{0}{\overset{r}{\int }}hx\left(1-\frac{x}{r}\right)dx}$

(iii) $2\pi {\displaystyle \underset{0}{\overset{r}{\int }}2x\sqrt{r^2-x^2}dx}$

(iv) $2\pi {\displaystyle \underset{0}{\overset{b}{\int }}2ax\sqrt{1-\frac{x^2}{b^2}}dx}$

(v) $2\pi {\displaystyle \underset{-r}{\overset{r}{\int }}2\left(R-x\right)\sqrt{r^2-x^2}dx}$

(d)

To determine

The correct option for the integral that describes the volume of a right circular cone along with the dimensions of the torus. The options are:

(i) $2\pi {\displaystyle \underset{0}{\overset{r}{\int }}hxdx}$

(ii) $2\pi {\displaystyle \underset{0}{\overset{r}{\int }}hx\left(1-\frac{x}{r}\right)dx}$

(iii) $2\pi {\displaystyle \underset{0}{\overset{r}{\int }}2x\sqrt{r^2-x^2}dx}$

(iv) $2\pi {\displaystyle \underset{0}{\overset{b}{\int }}2ax\sqrt{1-\frac{x^2}{b^2}}dx}$

(v) $2\pi {\displaystyle \underset{-r}{\overset{r}{\int }}2\left(R-x\right)\sqrt{r^2-x^2}dx}$

(e)

To determine

The correct option for the integral that describes the volume of a right circular cone along with the dimensions of the torus. The options are:

(i) $2\pi {\displaystyle \underset{0}{\overset{r}{\int }}hxdx}$

(ii) $2\pi {\displaystyle \underset{0}{\overset{r}{\int }}hx\left(1-\frac{x}{r}\right)dx}$

(iii) $2\pi {\displaystyle \underset{0}{\overset{r}{\int }}2x\sqrt{r^2-x^2}dx}$

(iv) $2\pi {\displaystyle \underset{0}{\overset{b}{\int }}2ax\sqrt{1-\frac{x^2}{b^2}}dx}$

(v) $2\pi {\displaystyle \underset{-r}{\overset{r}{\int }}2\left(R-x\right)\sqrt{r^2-x^2}dx}$