How to find the volume of a square with function-based side?

[Eclair_de_XII]

Homework Statement

"[Find the volume of a] solid whose base is the region bounded by the curves (y = x²) and y = 2 - x² and whose cross sections through the solid perpendicular to the x-axis are squares."

Homework Equations

A(f(x)) = f(x)²
V = ∫(A(f(x))dx
Image of problem: http://i.imgur.com/6FtsBzO.png
Answer (from back of book): V = 64/15.

The Attempt at a Solution

y = (2 - x²) - (x²) = 2 - 2x²
A(y) = (2 - 2x²)² = 4x⁴ - 4x² + 4
V = ∫A(f(x)) = (4/5)x⁵ - (4/3)x³ + 4x on [-1,1]
[(4/5) - (4/3) + 4] = [(12 - 20 + 60)/(15)] - [(-12 + 20 - 60)/(15)] = (104/15) ≠ (64/15)

I think I'm doing this right, but I'm not getting the right numbers. Could someone help me with this?

 

[Student100]

Homework Statement

"[Find the volume of a] solid whose base is the region bounded by the curves (y = x²) and y = 2 - x² and whose cross sections through the solid perpendicular to the x-axis are squares."

Homework Equations

A(f(x)) = f(x)²
V = ∫(A(f(x))dx
Image of problem: http://i.imgur.com/6FtsBzO.png
Answer (from back of book): V = 64/15.

The Attempt at a Solution

y = (2 - x²) - (x²) = 2 - 2x²
A(y) = (2 - 2x²)² = 4x⁴ - 4x² + 4
V = ∫A(f(x)) = (4/5)x⁵ - (4/3)x³ + 4x on [-1,1]
[(4/5) - (4/3) + 4] = [(12 - 20 + 60)/(15)] - [(-12 + 20 - 60)/(15)] = (104/15) ≠ (64/15)

I think I'm doing this right, but I'm not getting the right numbers. Could someone help me with this?

Check your setup math, (x^2-(2-x^2))^2 would be the region you want to look at.

You have two fours, I think you just made a math mistake. =)

 

[Eclair_de_XII]

You're right.

A(y) = (2 - 2x²)² = 4x⁴ - 8x² + 4
V = ∫A(f(x)) = (4/5)x⁵ - (8/3)x³ + 4x on [-1,1]
[(4/5) - (8/3) + (4)] - [(-4/5) + (8/3) - (4)] = 2(12 - 40 + 60)/(15) = 2(32/15) = 64/15

Thanks!