Find the average value of f over the given rectangle. f(x, y) = 4x2y, R has vertices (−1, 0), (−1, 4), (1, 4), (1, 0)

average-value

#1

Find the average value of f over the given rectangle. f(x, y) = 4x2y, R has vertices (−1, 0), (−1, 4), (1, 4), (1, 0).

Answer:

Make a sketch of a rectangle in coordinate system.

Notice that this is a rectangle with sides 2 and 4 and they are parallel to coordinate axes.

To find average value of a function we are going to find the integral of this function and divide the integral by area. Integration is done over the rectangle.
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