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 \(x\) -axis. (Make your selection on the basis of a sketch of the solid and not by performing any calculations.) \(y=\arctan x, \quad y=0, \quad x=0, \quad x=1\) (a) 10 (b) \(\frac{3}{4}\) (c) 5 (d) -6 (e) 15

Short answer

(b) \(\frac{3}{4}\) is the most reasonable answer, as it is the only option which is less than \(\frac{1}{3}\pi\), roughly equal to \(1.05\).

Step-by-step solution

Sketch the Boundaries

Sketch the boundaries of the solid by drawing the graph of \(y = \arctan x\) between \(x = 0\) and \(x = 1\). Include also the lines \(x = 0\), \(x = 1\), and \(y = 0\) in the graph.

Visualize the Solid

Envision the solid which is generated by revolving the region enclosed by the drawn lines about the x-axis. The shape should resemble a cone, with the base radius equal to \(\arctan 1\) or the maximum value of \(y\) from Step 1, and a height of \(1\) (from \(x = 0\) to \(x = 1\).

Estimate the Volume

Remember the formula for the volume of a cone \(V = \frac{1}{3}\pi r^2h\), where \(r\) is the radius and \(h\) is the height. The height is \(1\) and radius is less than \(1\), so the volume of our solid of revolution should be less than \(\frac{1}{3}\pi\). Inspect the given options and find the one closest to this estimate.