Final answer:
To calculate the double integral over the bounded region D between y=x² and x=y², determine the intersection points, set up the integral, compute the inner integral with respect to y, and then the outer integral with respect to x to find the area of D.
Step-by-step explanation:
To compute the double integral ∬D dA where D is the bounded region between the curves y=x² and x=y², we need to find the points of intersection of the two curves and set up the integral. We can solve for the intersection points by setting x=y² and y=x² equal to each other. This gives us x³=x, suggesting the solutions x=0 and x=±1.
Visually, this creates two regions, but due to the symmetry, we can focus on the upper right region where x and y are positive and multiply the integral by 2 to account for the lower left region. The boundary curves crossover at the points (0,0) and (1,1). Thus, we integrate y=x² with respect to y from 0 to x and then x with respect to x from 0 to 1. The integral can be expressed as:
∬∬D dA = 2 ∠₁₀ ∠⁴⁰ (x² - y²) dy dx
The first step is to perform the inner integral with respect to y, then the outer integral with respect to x. This gives us the area of the region D.