Math help please !!

Question

Consider the region in the first quadrant bounded by the line 8x − y = 0

and the curve 8x^³ − y = 0.

 Find the volume of the solid of revolution generated by revolving this region about the line x = −1

 using the disk/washer method. 

Answer 1

Hi Diana,

It's hard to explain this problem without a well drawn sketch of the region. Since we want to rotate the region bounded by the line and the curve about the line x=-1 we will have make sure we measure the washer dimensions correctly.

The volume element of the washer is dV=(πx_c² -πx_L²) where c_c is the distance from the line x=-1 to the curve. x_c=(y^1/3/2)+1 and x_L=(y/8)+1

Then the volume V=∫(πx_c² -πx_L²)dy=2π∫(y^1/3/2)+1)²-(y/8)+1)²dy where the limits are y=0 to y=8 ( just calculated 1/2 the volume and doubled it).

Can you take it from there? Let me know if you have questions

Regard

Jim