How do you find the volume of a solid that is enclosed by #y=secx#, #x=pi/4#, and the axis revolved about the x axis?

Question

Scarlett Allison · Accepted Answer

Note that #secxgt=1# for all values of #x#.

Here's the graph of #secx#:

graph{secx [-14.24, 14.24, -7.12, 7.12]}

There is no region bounded by #secx#, #x=pi//4# and the axis without another vertical line in the form #x=a# to serve as a boundary of our region.

We don't have enough information to complete this problem.

Ezra Adams · Answer

I have solved this way:

Christopher Allers · Answer

undefined To find the volume of the solid generated by revolving the region enclosed by the curve $ y = \sec(x) $, the line $ x = \frac{\pi}{4} $, and the x-axis about the x-axis, you would use the method of cylindrical shells.

1. Determine the limits of integration by finding the points of intersection between $ y = \sec(x) $ and $ x = \frac{\pi}{4} $.

2. Set up the integral for the volume using the formula for cylindrical shells:

$$ V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx $$

where $ f(x) $ is the function representing the curve, and $ a $ and $ b $ are the limits of integration.

3. Integrate the expression obtained in step 2 with respect to $ x $ over the interval determined in step 1.

4. Evaluate the integral to find the volume of the solid.