Question

Find the average value of f over the given rectangle f (x, y) =

2x2y, R has vertices (−3, 0), (−3, 3), (3, 3), (3, 0)

Answer 1

To find the average value of the function f(x, y) = 2x2y over the rectangle with vertices (-3, 0), (-3, 3), (3, 3), (3, 0), compute the double integral of f(x, y) over the region and divide by the rectangle's area.

Step-by-step explanation:

The average value of a function f(x, y) over a rectangle can be found using the double integral over the region R. The rectangle R given by the vertices (−3, 0), (−3, 3), (3, 3), (3, 0) can be described by the limits −3 ≤ x ≤ 3 and 0 ≤ y ≤ 3. The function f(x, y) = 2x2y is defined over this rectangle.

To find the average value of f over R, we integrate f with respect to x and y over these limits and divide by the area of R. The steps are as follows:

1. Calculate the area of R, which is (3 – (−3)) × (3 – 0) = 6 × 3 = 18.
2. Set up the double integral of f(x, y) over R: ∫∫_R 2x2y dA.
3. Integrate with respect to y from 0 to 3, then integrate the resulting expression with respect to x from −3 to 3.
4. Calculate the integral and divide by the area of R to get the average value.

By following these steps, we can calculate the average value of the function f(x, y) over the region R.