Question

Use the integration capabilities of a graphing utility to approximate the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=\sqrt{2 x}, \quad y=x^{2} $$

Short answer

To find the volume of the solid, graph the functions, find the intersection points along the \(x\)-axis, apply the disk method of integration using the difference of the functions as the radial function, and integrate over the range defined by the intersection points.

Step-by-step solution

Plot the Functions

Enter the two functions \(y=\sqrt{2x}\) and \(y=x^{2}\) into a graphing utility to plot their graphs.

Find Intersection Points

Find the points where the two functions intersect. This can be done by setting the two functions equal to one another as follows: \(\sqrt{2x} = x^{2}\)Solving this equation provides the intersection points along the \(x\)-axis, which will serve as the bounds of the integral.

Apply Disk Method

Use the disk method formula for finding the volume of a solid of revolution, which is \( V = \pi \int_{a}^{b} [f(x)]^{2} dx\). The radial function \(f(x)\) is the distance from the surface of the solid to the axis of rotation at any given \(x\), which in this case is the difference between the two original functions, \(\sqrt{2x}\) and \(x^{2}\). So the formula becomes \( V = \pi \int_{a}^{b} [\sqrt{2x} - (x^{2})]^{2} dx\).

Perform the Integration

Integrate the resulting function from the previous step over the interval \([a, b]\), which is obtained from the intersection points. This operation can be done with a graphing calculator or an online integration tool.