Final answer:
To set up the iterated integrals, we need to determine the bounds of integration for both x and y. The given conditions for D are y = x - 6 and x = y^2. We can find the bounds by setting these equations equal to each other. After finding the bounds, we can evaluate the double integral by choosing the easier order of integration and substituting the values into the integrand.
Step-by-step explanation:
To set up the iterated integrals, we need to determine the bounds of integration for both x and y. The given conditions for D are y = x - 6 and x = y^2. We can find the bounds by setting these equations equal to each other:
x - 6 = x^2
Then rearrange the equation:
x^2 - x + 6 = 0
Using the quadratic formula, we find:
x = (1 ± √(1 - 4(6)))/2
Since x cannot be negative in this case, we only consider the positive root:
x = (1 + √23)/2
Therefore, the bounds of x are 0 and (1 + √23)/2.
Next, we need to determine the bounds of y. Since x = y^2, we substitute this equation into y = x - 6:
y = y^2 - 6
Rearrange the equation:
y^2 - y + 6 = 0
Using the quadratic formula again, we find:
y = (1 ± √(1 - 4(6)))/2
Since y cannot be negative in this case, we only consider the positive root:
y = (1 + √23)/2
Therefore, the bounds of y are 0 and (1 + √23)/2.
Evaluating the double integral:
Since we need to choose the easier order of integration, we will evaluate the integral in terms of y first. The bounds for y are 0 to (1 + √23)/2. The integrand is (7x - 3y), so we substitute x = y^2 into the integrand:
(7(y^2) - 3y)
Now we can evaluate the integral:
∫((7(y^2) - 3y)dy)
After evaluating the integral, the final result will be the value of the double integral.