Solve the problem.
-A region in the first quadrant is bounded above by the curve $y = \cosh x$, below by the curve $y = \sinh x$, on the left by the $y$-axis, and on the right by the line $x = 12$. Find the volume of the solid generated by revolving the region about the $x$-axis.
A) 0
B) $\frac { \pi } { 2 } \left( e ^ { - 24 } + 1 \right)$
C) $12 \pi$
D) $2 \pi$

Question

Quizplus · Accepted Answer

We start by trying to find the points of intersection between the two curves. Setting them equal, we get $\cosh x=\sinh x$, or $\tanh x=1$, which yields $x=\operatorname{arctanh} 1=\infty$. Since that is not a valid solution, we must have that the two curves do not intersect.

The two curves both intersect the origin, so that is the point of our solid. To find the other boundaries of the solid, we need to find the $x$-coordinate of the point on $\cosh x$ that is closest to $x=12$, and the $x$-coordinate of the point on $\sinh x$ that is farthest from $x=0$. The first point can be found using $\cosh x \ge e^x/2$ for all $x$, and so we want the value of $x$ that satisfies $$\frac{e^{12}}{2}=\cosh x=\frac{e^x+e^{-x}}{2}.$$Solving this yields $e^x=15$, or $x=\ln 15$. The second point can be found similarly by observing that $\sinh x \le e^x/2$ for all $x$, and so we want the value of $x$ that satisfies $$\frac{e^x}{2}=\sinh x=\frac{e^x-e^{-x}}{2}.$$Solving this yields $x=\ln 3$.

Thus, the solid is a region of revolution, and its volume can be computed with the formula $$\int_a^b \pi y^2\,dx$$where $y$ is the distance from the curve to the $x$-axis. The two curves don't intersect, so we can choose the limits of $a=0$ and $b=12$ without having to split up the integral.

Thus the volume of the solid is \begin{align*}
\int_0^{12}\pi\left(\cosh x\right)^2\, dx-\int_0^{12}\pi(\sinh x)^2\,dx &=\pi\int_0^{12}\left(\cosh^2 x-\sinh^2 x\right)\,dx \
&=\pi\int_0^{12}\cosh(2x)\,dx \
&=\frac{\pi}{2}\left[\sinh(24)\right] \
&=\boxed{12\pi}.
\end{align*}

Solve the Problem y=cosh⁡xy = \cosh xy=coshx , Below by the Curve

Solve the Problem $y = \cosh x$ , Below by the Curve