Final answer:
To evaluate the given double integral, set up the limits of integration correctly. Simplify the function and evaluate the inner integral with respect to x. Then evaluate the outer integral with respect to y to obtain the final result.
Step-by-step explanation:
To evaluate the double integral of the function y² / √x² + y² over the given region, we need to set up the limits of integration correctly. The inner integral is with respect to x, and the limits of integration for x are from 0 to 21−y². The outer integral is with respect to y, and the limits of integration for y are from 0 to 1/√2.
We can start by evaluating the inner integral. Simplify the expression y² / √x² + y² by multiplying the numerator and denominator by √(x² + y²): y² / √x² + y² = y²√(x² + y²) / (x² + y²). Now integrate this expression with respect to x, using the limits of integration 0 to 21−y².
After evaluating the inner integral, we obtain an expression in terms of y. Now we need to evaluate the outer integral, which is ∫ [expression in terms of y] dy, using the limits of integration 0 to 1/√2. Evaluate this integral to get the final result.