Question

Determine which value best approximates the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(y\) -axis. (Make your selection on the basis of a sketch of the solid and not by performing any calculations.)\(y=\tan x, y=0, x=0, x=\frac{\pi}{4}\) (a) 3.5 (b) \(-\frac{9}{4}\) (c) 8 (d) 10 (e) 1

Short answer

The best approximate value for the volume of the solid generated by revolving the bounded region about the y-axis is (a) 3.5.

Step-by-step solution

Sketch the given equations

Start by sketching the functions \(y=\tan x\), \(y=0\), \(x=0\) and \(x=\frac{\pi}{4}\) on a coordinate plane. The sketch would show that \(y=\tan x\) raises from 0 to \(\infty\) between \(x=0\) and \(x=\frac{\pi}{4}\), \(y=0\) is a straight line on the x-axis, \(x=0\) is a straight line on the y-axis, and \(x=\frac{\pi}{4}\) is a vertical line crossing the y-axis at \(\frac{\pi}{4}\). The bounded region is a triangle.

Visualize the solid

Next, imagine the sketched region being revolved around the y-axis. It would generate a cone-shaped solid.

Approximate the volume

Visually, it can be seen that when the solid is formed by revolving around the y-axis, it forms a cone shape. As no calculations are performed, the volume could be approximated fairly small, not as dense as a full cylinder (pi) but smaller than half of pie. The closest approximate value from the given options would be option (a) 3.5.