Find out the volume of the solid obtained by rotating the region bounded by x =  (y - 2)² and  y = x around the line y = -1.

Solution : We have to first get the intersection points there.

y =( y - 2)²

y = y ² - 4 y + 4

0 =y ² - 5 y + 4

0 = ( y - 4) ( y -1)

Therefore, the two curves will intersect at y = 1 & y = 4 . Following is a sketch of the bounded region and the solid.

Following are our sketches of a typical cylinder. The sketch on the left is here to illustrates some context for the sketch on the right.

Following is the cross sectional area for this cylinder.

A ( y ) = 2 ∏ ( radius ) ( width )

=2 ∏ ( y + 1) ( y - ( y - 2)² )

= 2 ∏ (- y³ + 4 y ² + y - 4)

The first cylinder will cut in the solid at y = 1 and the final cylinder will cut in at y = 4 . Then the volume is

V=∫^d_c A(y)dy

=2 ∏∫⁴₁  -y³ + 4y²  + y- 4 dy

=2 ∏ (-(1/4) y⁴ + (4/3)y³+(1/2)y² - 4y)|⁴₁

=(63 ∏/2)