Integrals? Please help

Question

Find the area of the region R bounded by the graphs of the equations y-y^3=x and x=0.

Suppose you rotate R around the y-axis. find the volume of the resulting solid.

Answer 1

Set x = 0 to find the y-intercepts: y(1 - y²) = 0

 

y = 0, -1, or 1.

 

The function is odd, so the net area over symmetric limits is zero.  But if we want to compute the total area to the left and right of the y-axis we have make the negative area positive:

 

 

              _{0                               1}

Area: |∫(y - y³) dy| + |∫(y - y³)| = _____

              ^{-1                             0}

 

 

 

We can calculate the volume by integrating disks stacked along the y-axis: dV = πr² dy. 

 

Since r = x, dV = πx² dy.

 

                 ₁

Volume: ∫π(y - y³)² dy = _____

                 ^-1

 

 

 

Ans: A = 1/2; V = 16π/105