HELP - CYLINDRICAL SHELL METHOD CALCULUS

Question

By using the cylindrical shell method, find the volumes of the solids generated by revolving the regions bounded by the given lines and the curves about the y-axis. (So the axis of revolution for both parts in this question is the y-axis).

a. y=x^2 , y=2-x , x≥ 0

b. y=2-x^2 , y= x^2 , x≥ 0

*for both a and b, x is greater than or equal to 0.

Answer 1

First notice that

2 – x > x² = (x + 1/2)² – x – 1/4

for all

0 ≤ x < √(2 + 1/4) – 1/2 = 1.

So the volume is

v = ∫₀¹a(x)dx,

where the area of the cylindrical shell of radius x and thickness dx is

a(x) = 2πxh(x)

and its height is

h(x) = 2–x–x² > 0.