Calculate the double integral of f(x, y) = 100 - 6x²y dA for the region R: 0 ≤ x ≤ 2, -1 ≤ y ≤ 1.

Steps to Solve

1. Set up the double integral

We will set up the double integral for the function $f(x, y) = 100 - 6x^2y$ over the region $R$ defined by $0 \leq x \leq 2$ and $-1 \leq y \leq 1$:

$$ \iint_R f(x, y) , dA = \int_{0}^{2} \int_{-1}^{1} (100 - 6x^2y) , dy , dx $$

2. Evaluate the inner integral

Next, we will evaluate the inner integral with respect to $y$:

$$ \int_{-1}^{1} (100 - 6x^2y) , dy $$

Calculating it:

• The integral of $100$ with respect to $y$:

$$ 100y \bigg|_{-1}^{1} = 100(1) - 100(-1) = 100 + 100 = 200 $$

• The integral of $-6x^2y$ with respect to $y$:

$$ -6x^2 \frac{y^2}{2} \bigg|_{-1}^{1} = -3x^2(1^2) - (-3x^2(-1)^2) = -3x^2 - (-3x^2) = 0 $$

Thus, the result of the inner integral is:

$$ \int_{-1}^{1} (100 - 6x^2y) , dy = 200 $$

3. Evaluate the outer integral

Now we substitute the result from the inner integral into the outer integral:

$$ \int_{0}^{2} 200 , dx $$

Calculating it:

$$ 200x \bigg|_{0}^{2} = 200(2) - 200(0) = 400 $$