How to find the volume of the solid of revolution?

Question

Q: Find the volume of the solid of revolution obtained by revolving the region bounded by the line y = 2 and the curve y = sec^2x, -90 < x < 90, around the x-axis.

Attempted solution: 1/3 tanx ( sec^2(x) + 2)

I'm sure I did the integral correctly, but now when I need to sub in for 90, tan(90) does not exist? (The answer is 2(pi)^2 - 8pi/3)

Answer 1

Presumably your arguments are degrees, not radians. The volume element of the solid of revolution formed by the rectangle of width $dx$ and extending from $y_1$ to $y_2$ is $\pi(y_2^2-y_1^2)dx$ You integrated $\sec^4(x)dx$ correctly, but that is not quite what you need to integrate.

$\sec^2(x)$ crosses 2-are you supposed to just consider the part of it below y=2? This will eliminate the problem of the tangent blowing up. If you are supposed to consider the area outboard of y=2 all the way to 90 degrees, then the volume is infinite, as your calculation shows.