Final answer:
To calculate the iterated integral ∦∦(6x²y - 2x) dy dx over the given region, first integrate with respect to y, then with respect to x. The bounds for x and y are 0 to 3 and 1 to 2 respectively.
Step-by-step explanation:
The student has asked to calculate the iterated integral of ∦∦(6x²y - 2x) dy dx over the region R, bounded by x = 0, x = 3, y = 1, and y = 2. To solve this, we will evaluate the integral with respect to y first, keeping x constant, and then integrate the resulting expression with respect to x.
First, we integrate with respect to y:
∫ (6x²y - 2x) dy, between y = 1 and y = 2.
We find the antiderivative:
3x²y² - 2xy | ₁² = (3x²(2)² - 2x(2)) - (3x²(1)² - 2x(1))
Simplify and solve for x from 0 to 3:
12x² - 4x - (3x² - 2x) = 9x² - 2x
Now, integrate with respect to x:
∫ (9x² - 2x) dx, between x = 0 and x = 3.
The result of this second integral gives us the final answer for the iterated integral over the region R.