Chapter A5.1, Problem 43E

To determine

To find: the volume of the solid.

Answer to Problem 43E

The volume of the solid is $V=24\pi$ .

Explanation of Solution

Given:

The line $x=4,$ and the hyperbola $x^2-y^2=1$ .

Calculation:

Equation of hyperbola: $x^2-y^2=1$ .

This implies that,

  $x^2=y^2+1$

  $x=\pm \sqrt{y^2+1}$

Since $y$ cannot be negative, use the positive root on the interval $-3\le y\le 3$ .

  $V={\displaystyle {\int }_{-3}^3\pi }{\left(\sqrt{y^2+1}\right)}^2\cdot dy$

  $=2{\displaystyle {\int }_0^3\pi }\left(y^2+1\right)\cdot dy$

  $=2\pi {\left[\frac{1}{3}{y}^3+y\right]}_0^3$

  $=24\pi$

Therefore, $V=24\pi$ .

Conclusion:

Therefore, the volume of the solid is $V=24\pi$ .