Math GRE - #28

What is the volume of the solid formed by revolving about the x-axis the region in the first quadrant of the xy-plane bounded by the coordinate axes and the graph of the equation: $$y=\frac{1}{\sqrt{1+x^2}}?$$

1. $\dfrac{\pi}{2}$
2. $\pi$
3. $\dfrac{\pi^2}{4}$
4. $\dfrac{\pi^2}{2}$
5. $\infty$

Solution :

Choice 4 is the correct answer.

Recall that the volume of revolution around the x-axis is given by the formula $$V=\int{\pi y(x)^2}\,dx.$$ In this case, we can substitute for $y$ and plug in the limits of integration to obtain: $$V=\pi\int_0^\infty{\frac{1}{1+x^2}}\,dx.$$
Almost everyone with some calculus experience has had the great displeasure of memorizing: $$\int{\frac{1}{1+x^2}}\,dx = \arctan x + C.$$ Now we can put it to good use! We can see that the answer is (if you'll pardon my poor notation): $$V=\pi\int_0^\infty{\frac{1}{1+x^2}}\,dx=\pi\left[\arctan\infty-\arctan0\right]=\pi\cdot\frac{\pi}{2}=\frac{\pi^2}{2}.$$

Interesting tidbit: Here's a funny little thing you'll often see in first year calculus. It's called Gabriel's Horn. It's got finite volume but INFINITE surface area. Just something interesting I wanted to share.