Question

Evaluate the iterated integral. ∫[0 to 1] ∫[x^2 to x] (6+12y) dy dx

Answer 1

To solve the iterated integral, one should first integrate (6+12y) with respect to y, then integrate the resulting expression with respect to x.

Step-by-step explanation:

To evaluate the iterated integral ∫[0 to 1] ∫[x^2 to x] (6+12y) dy dx, we need to perform the integration in two steps. First, we integrate with respect to y, then with respect to x.

1. Integrate (6+12y) with respect to y from y = x^2 to y = x. This will give us a function of x.
2. Integrate the resulting function of x with respect to x from x = 0 to x = 1.

Let's perform the first step of the calculation:

∫[x^2 to x] (6+12y) dy = [6y + 6y^2]_x^2_x = 6x - 6x^2 + 6x^3 - 6x^4

Now, we integrate the resulting expression with respect to x:

∫[0 to 1] (6x - 6x^2 + 6x^3 - 6x^4) dx

By computing this final integral, we'd obtain the value of the iterated integral.