Question

In Exercises \(21-24,\) use the shell method to find the volume of the solid generated by revolving the plane region about the given line. $$ y=x^{2}, \quad y=4 x-x^{2}, \text { about the line } x=4 $$

Short answer

The volume of the solid, evaluated by the shell method, is found by evaluating the integral from steps 3 and 4.

Step-by-step solution

Identify the curves and the axis of revolution

The curves are given by the equations \(y = x^{2}\) and \(y = 4x - x^{2}\), and they are revolved around the line \(x=4\). The area between these curves will generate the volume of the solid when rotated.

Find points of intersection

To find out the limits of integration, set \(x^{2} = 4x - x^{2}\). Solving it gives \(x = 0, 2\). These are the boundaries of the region being rotated.

Set up the shell method integral

For the shell method, the volume \(V\) can be found using the integral formula \(V = 2\pi \int_{a}^{b} r(x) h(x) dx\), where \(r(x)\) is the distance from the axis of revolution to the centroid of the shell and \(h(x)\) is the height of the shell. From the diagram, \(r(x) = 4 - x\) and the height of each shell is the difference in \(y\)-values of the two functions, \(h(x) = 4x - x^{2} - x^{2}\). So the volume is \(V = 2\pi \int_{0}^{2} (4 - x)(2x^{2} - 4x) dx\).

Evaluate the Integral

Evaluating the integral \(V = 2\pi \int_{0}^{2} (8x^{2} - 12x^{3} + x^{4}) dx\) gives the final volume of the solid.