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

Steps to Solve

1. Set up the double integral

We begin by expressing the double integral of the function over the given region ( R ) using the bounds provided. The double integral can be written as:

$$ \iint_R (100 - 6x^2y) , dA = \int_0^2 \int_{-1}^1 (100 - 6x^2y) , dy , dx $$

2. Integrate with respect to ( y )

We first perform the inner integral with respect to ( y ):

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

This can be split into two parts:

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

Calculating these:

• For the first integral:

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

• For the second integral:

The integral of an odd function over a symmetric interval results in zero:

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

Thus, the result of the inner integral becomes:

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

3. Integrate with respect to ( x )

Now we substitute the result of the inner integral back into the outer integral:

$$ \int_0^2 200 , dx $$

Calculating this:

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