Volume integral

Question

Find the volume of the solid obtained when the region bounded by y=√x ,y=-x and the line x=9 is revolved around the x axis.

Answer 1

This problems is a bit tricky because the resulting solid should be viewed as being into two parts joined together.

 

The first part corresponds to 0 ≤ x < 1 and the second to   1 ≤ x  ≤ 9

 

The volume of the first part can be computed using the disk method   :  The integration is over x with limits 0 and 1

  the integrand  is  π x    leading to V1 =  π/ 2 

 

The second can also be computed using the disk method.   The integration is over x with limits 1 and 9

   the integrand is   π x²       leading to V2 = π/3 (9³ -1 )

 

The total volume is  V1 + V2 =  π [ 1/2 + 81 - 1/3)