Shell Method: Between x^2 and 4x - x^2 around y-axis

Shell Method

Volume of the space between y = x^2 and y = 4x – x^2 when it is revolved around the  y axis.

 

First of all, draw a picture.

 

The shaded area is the area we're interested in.

We ask some questions to organize our thought process.                                                                                                                                                              

1.       What is the area of the rectangle? – The width of the rectangle is dx, the height will be 4x – x² –x², or 4x  − 2x² (The higher curve minus the lower curve.)

2.       What is the radius?- With the shell method, the radius is the distance between the rectangle and the line we're revolving around. In this case, it will be x.

3.       Will our integral be in terms of x or y? – the integral will be in terms of x, because the radius is along the x. (With the shell method, if we're revolving around a vertical line, everything will be x, if around a horizontal line, everything will be y.)

4.       What are our boundaries of x? – We will integrate between 0 and 2 (See diagram, or set x² equal to 4x – x².

What we want is 2pi times the integral of the area of the rectangle times the radius. In this case it will be:

2pi times the integral of( x times 4x minus 2x² times dx )= 2pi times (8/3) = 16pi/3.