Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis.Verify your results using the integration capabilities of a graphing utility. , , ,

Question

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis.Verify your results using the integration capabilities of a graphing utility. ​  ,  ,  ,  ​

Quizplus · Accepted Answer

The region bounded by the given equations is shown in the figure below:

![Graph1](https://i.imgur.com/ybI9Kn2.png)

To find the volume of the solid generated by revolving this region about the x-axis, we need to use the method of cylindrical shells. We can imagine dividing the region into thin vertical strips, each of width dx. Revolving each of these strips about the x-axis generates a cylindrical shell of thickness dx and radius x, as shown in the figure below:

![Graph2](https://i.imgur.com/wZFgvdh.png)

The volume of each cylindrical shell is given by:

dV = 2πx(y2-y1)dx

Since the region is symmetric about the y-axis, we can compute the volume by integrating the above expression from x=0 to x=1. To do this, we need to express y2 and y1 in terms of x. From the given equations, we have:
y2 = √x and y1 = 1-x

Substituting these values in the above expression, we get:

dV = 2πx(√x - (1-x))dx

Simplifying, we get:

dV = 2π(x3/2 - x2 + x)dx

Integrating both sides from x=0 to x=1, we get:

V = ∫0^1 2π(x3/2 - x2 + x)dx

= 2π(2/5 - 1/3 + 1/2)

= 2π(16/30)

= 32π/15

Using a graphing utility, we can verify that the volume is indeed 32π/15. Hence, the correct answer is (C).

Find the Volume of the Solid Generated by Revolving the Region