Calculating Volume. Applications of Integration.

Question

I've been working on the same problem for a bit. I can't figure out what the volume of the problem is. One of my answers is pi over 3, but it doesn't seem right. Would you be able to confirm this?

The problem is x=y^2 and x=1-y^2; about x=3.

Answer 1

f(y) = y² g(y) = 1-y²

Setting f(y) = g(y)

y²=1-y² --> y = ±√1/2 --> a = -√1/2 b= √1/2

Volume, V = π∫_a^b|(f(y)-d)²-(g(y)-d)²|dy where d = 3 in this case

V = π∫_a^b|(y²-3)²-(-y²-2)²| dy = π∫_a^b|-10y²+5|dy = π(-10y³/3 +5y)|_a^b = π(-5√2/3+5√2) = (10/3)√2π