Question

Evaluate the double integral. $$ \int_{0}^{2} \int_{3 y^{2}-6 y}^{2 y-y^{2}} 3 y d x d y $$

Short answer

The value of the double integral is 12.

Step-by-step solution

Integrate with respect to x

The integral of \(3y\) with respect to \(x\) is \(3xy\).

Apply limits of x

By replacing \(x\) by the limits of \(x\), the integral becomes \(3y \cdot (2y - y^2 - (3y^2 - 6y)) = -3y^3 + 9y^2\).

Integrate with respect to y

The integral of \(-3y^3 + 9y^2\) with respect to \(y\) is \(-3/4 * y^4 + 3y^3\).

Apply limits of y

By replacing \(y\) by the limits of \(y\), the answer becomes \(-3/4 * 2^4 + 3 * 2^3 - 0 = -12 + 24 = 12\).