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

The double integral we need to evaluate is given by:

$$ \iint_R (100 - 6x^2y) , dA $$

with the region ( R ) defined by ( 0 \leq x \leq 2 ) and ( -1 \leq y \leq 1 ).

2. Iterate the integral

We will integrate first with respect to ( y ) and then with respect to ( x ):

$$ \int_{0}^{2} \left( \int_{-1}^{1} (100 - 6x^2y) , dy \right) dx $$

3. Evaluate the inner integral

Calculate the inner integral:

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

This will involve finding the integral of each term separately.

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

4. Compute the first part

Calculating the first integral:

$$ \int_{-1}^{1} 100 , dy = 100[y]_{-1}^{1} = 100(1 - (-1)) = 100(2) = 200 $$

5. Compute the second part

Now for the second integral:

$$ \int_{-1}^{1} 6x^2y , dy = 6x^2 \left[ \frac{y^2}{2} \right]_{-1}^{1} = 6x^2 \left( \frac{(1)^2}{2} - \frac{(-1)^2}{2} \right) = 6x^2(0) = 0 $$

Thus, the inner integral becomes:

$$ \int_{-1}^{1} (100 - 6x^2y) , dy = 200 - 0 = 200 $$

6. Evaluate the outer integral

Now substitute into the outer integral:

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

Calculating this integral gives:

$$ 200[x]_{0}^{2} = 200(2 - 0) = 400 $$