Find the Volume

Question

Use your calculator to find the approximate volume in cubic units of the solid created when the region under the curve y = sec(x) on the interval [0,] is rotated around the x-axis.

Question options:

1)	1.317
2)	1.732
3)	4.137
4)	5.441

Answer 1

The correct answer is #4).

To solve this problem, visualize a cross section of the solid perpindicular to the x-axis at x value x_0. Because we are rotating about the x-axis, the cross section is a disc of radius x₀. The area of the disc is π*sec²(x₀). To compute volume, integrate this area from x = 0 to x = π÷3: π*∫ sec²(x) dx. The integral of sec²(x) is tan(x). So the volume is π*(tan(π/3) - tan(0)) = π*√3.