Question

In Exercises \(35-38,\) (a) use a graphing utility to graph the plane region bounded by the graphs of the equations, and (b) use the integration capabilities of the graphing utility to approximate the volume of the solid generated by revolving the region about the \(y\) -axis. $$ x^{4 / 3}+y^{4 / 3}=1, x=0, y=0, \text { first quadrant } $$

Short answer

After performing the proper calculations using the graphing utility, one can obtain the approximate volume of the solid generated by revolving the region about the y-axis. Since the integral can't be solved analytically, the short answer would be the numerical evaluation of the integral from step 3.

Step-by-step solution

Graph the function

First, you should graph the function using a graphing utility like a graphing calculator or an online graphing tool. The function to be graphed is \(x^{4 / 3}+y^{4 / 3}=1\), and you should also graph the lines \(x=0\) and \(y=0\) to indicate the region of interest. Note that we're only interested in the first quadrant, so ignore any part of the graph that isn't in the first quadrant.

Set up the integral

To set up the integral to find the volume of rotation, first, we need to express \(x\) in terms of \(y\). From the given equation, \(x^{4/3} = 1 - y^{4/3}\). The volume \(V\) of the solid generated by revolving the region about the y-axis is given by \(V = \pi \int (x(y))^2 dy\), with \(x(y) = (1 - y^{4/3})^{3/4}\). Therefore, our integral will look like this: \(V = \pi \int_0^1 (1 - y^{4/3})^{3/2} dy\).

Compute the integral

Now, use the integral function of the graphing utility to approximate the volume of the solid. This can be done by programming the integral into the calculator and using the calculator's numerical approximation capabilities.