Find the volume of the solid between the explicit graph z = x(x-1)y(y-1) and the square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1?
Hello can someone work this problem out and explain to me how to do it. For my final answer, I got 0, so I think I'm doing it wrong. Thanks for all help in advance!
My work:
I simplified the equation to z = (x^2y^2-xy^2 - x^2y + xy)
Then took the integral dx, with respect to y and got (y^2/3 - y/3 - y^2/2 + y/2)
Then I took the integral dy of that, with respect to x and got 0.
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Verified answer
V = ∫(x = 0 to 1) ∫(y = 0 to 1) x(x-1) y(y-1) dy dx
...= ∫(x = 0 to 1) x(x-1) dx * ∫(y = 0 to 1) y(y-1) dy, since the bounds are constant
...= ∫(x = 0 to 1) (x^2 - x) dx * ∫(y = 0 to 1) (y^2 - y) dy
...= (x^3/3 - x^2/2) {for x = 0 to 1} * (y^3/3 - y^2/2) {for y = 0 to 1}
...= (-1/6) * (-1/6)
...= 1/36.
I hope this helps!