Question

In Exercises 21 and 22, find the volume of the solid generated by revolving the region about the given line. the region in the first quadrant bounded above by the line \(y=\sqrt{2},\) below by the curve \(y=\sec x \tan x,\) and on the left by the \(y\) -axis, about the line \(y=\sqrt{2}\)

Short answer

The volume of the solid of revolution is given by the integral \[2\pi \int_{0}^{π/4} ((\sqrt{2} - \sec x \tan x) \sec x \tan x)\, dx\] which might need a numerical approach for its evaluation.

Step-by-step solution

Determine the radius and height

The radius for cylindrical shells is the distance from the shell to the axis of revolution, which is \(y=\sqrt{2}-y\) in our case. The height function is the distance from the shell to the x-axis, represented by \(\sec x \tan x\). So, we get \(r(h)= \sqrt{2} - y\) and \(h'(x) = \sec x \tan x\).

Determine the Bounds of the Integral

The area in question is in the first quadrant and bounded by the \(y\)-axis on the left. This means, \(x\) ranges from \(0\) to \(π/4\). So, \(a = 0\) and \(b = π/4\).

Set up the Volume Integral

Substitute the radius, the height and the bounds into the volume formula to get the volume integral: \[V = 2\pi \int_{0}^{π/4} ((\sqrt{2} - \sec x \tan x) \sec x \tan x)\, dx\]

Evaluate the Integral

This integral is a bit tricky to evaluate. Using trigonometric identities to simplify, and then conducting integration, the exact result might not be straightforward to find. A numerical approximation using techniques like Riemann sums or numerical software can be used to evaluate the integral.