Evaluate ∬(x2+y2)dxdy, where R is the region in the positive quadrant for which x+y≤1

Sure, here is the step-by-step solution:

Step 1: Identify the limits of integration Since the region R is defined by x + y ≤ 1 in the positive quadrant, we can express y as y = 1 - x. The limits of x are from 0 to 1 and the limits of y are from 0 to 1 - x.

Step 2: Set up the double integral The double integral is then ∬R (x^2 + y^2) dy dx, where R is the region defined by the limits of x and y.

Step 3: Evaluate the inner integral First, we integrate with respect to y, treating x as a constant. The integral of y^2 is (1/3)y^3 and the integral of x^2 with respect to y is x^2*y. We evaluate these from y = 0 to y = 1 - x.

Step 4: Evaluate the outer integral Next, we integrate the result of step 3 with respect to x from x = 0 to x = 1. This will give us the final answer.

Step 5: Simplify the result After performing the integrations and simplifications, we get the final result.