Q: Find the volume generated by area bounded by two curves rotated through 360 degrees about the y-axis:

a) y = x²

y = 2- x²

b) y = x2

y2 = 8x

For these two parts, which formulae should I use?

Volume generated

= π ∫ x² dy

or = π ∫ [f(y)]² – [g(y)]² dy ?

For part a, it will be simplier if I use the first formula where I simplify the y before using the first formula:

y = 2 - x² - x²

y = 2 - 2x²

x = √ (2 - y)/2

After using the first formula, my answer is π.

However, if I use the second formula, I got diferent answer:

y = x²

x = √y

y = 2 - x²

x = √(2 - y)

V = π ∫ [f(y)]² – [g(y)]² dy

= π ∫ [√y]² - [√(2 - y)]² dy

= .....

= π (4 - 4)

= 0

For part b, it seems that I can't use the first formula, as after simplify,

y = √(8x) - x²

it's hard to change to make x the subject in terms of y.

Using the second formula for part b, I got the answer 24/7π.

So how do we know which formula to be used for which question? Because the question is involving volume generated by area bounded by two curves, shouldn't we be using the second formula only? Why using both formulas I get different answers? Please help me check if my answer is correct.

Many thanks in advance!! =)