Use symmetry to evaluate the double integral of $f(x,y)=xy$ over the region $R$, where $R$ is the disk $x^2+y^2\le 4$.

• A. $4\pi$
• B. $0$
• C. $8$
• D. $2\pi$

Answer: (B)

Verified step by step guidance

1. Step 1: Recognize that the region R is a disk centered at the origin with radius 2, as the inequality x² + y² ≤ 4 describes a circle with radius 2.
2. Step 2: Observe the function f(x, y) = x * y. This function is odd with respect to both x and y, meaning that flipping the sign of x or y will flip the sign of the function.
3. Step 3: Use symmetry properties of the disk. The disk is symmetric about the x-axis, y-axis, and the origin. For any odd function integrated over a symmetric region like this, the integral evaluates to 0 because the positive contributions cancel out the negative contributions.
4. Step 4: Set up the double integral ∬_R x * y dA, where dA represents the area element. However, due to symmetry, you can conclude that the integral is 0 without performing explicit calculations.
5. Step 5: Conclude that the integral evaluates to 0 based on the symmetry argument, as the function x * y is odd and the region R is symmetric.