# How do you use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region # y = x^3#, #y = 0#, #x = 2# rotated about the x axis?

##### 1 Answer

The answer is

Firstly, you should graph your three functions to see clearly what is the delimited region :

We are rotating our plane region around the x-axis, which means we have to write our function

The radius of any cylinder in our final volume will be given by

And the width will be

(Look here for more details or diagrams about the cylinders)

Therefore, the cross-sectional area of any cylinder will be :

The radius of the cylinders, according to the delimited region, goes from :

So the volume of our solid, which is the sum of all the cross-sectional area of the cylinders, is :

The antiderivative of

Thus, the antiderivative of

Now we can calculate the integral :