Evaluate the triple integral ∭E(x+2y)dV where E is bounded by the parabolic cylinder y=5x2 and the planes z=x, y=5x, and z=0.

Question

I'm just confused about what the bounds are for when you integrate with respect to x, I thought it was 0 and 1, but I keep getting the wrong answer so I don't know anymore.

Answer 1

The order of integration matters! In a triple integral, the inner integral can have bounds expressed in terms of two variables, the middle integral in terms of one variable, and the outer integral can only have numbers as bounds. I tried setting up the integral like this: ∫0 to 1 ∫ 0 to x ∫ 5x^2 to 5x of (x +2y) dy dz dx

The boundaries on x can be zero to one, but that means x has to be the last variable of integration. The inner two integrals have boundaries in terms of x, and all of those are evaluated after integration. I hope this helps!