Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the y-axis. y = 4 - x

Question

 this is a problem that was given in class but i'm having trouble resolve it ..please help if you can

Answer 1

The plane region is a right triangle with vertices (0,4), (4,0), and (0, 0).  The right angle has vertex (0,0).

 

At a distance of x from the y-axis, take a thin vertical strip (of width dx) of the region (with the shell method, we take cross sections parallel to the axis of rotation) and rotate it about the y-axis to obtain a shell of radius x, height 4-x, and thickness dx.

 

The volume of the shell is 2πx(4-x)dx = 2π(4x - x²)dx 

 

Integrate from 0 to 4 to get the volume of the solid:

 

Volume = 2π(2x² - (1/3)x³) (from 0 to 4) = 2π(32 - 64/3) = 64π/3