Find the average value of f over the given rectangle. f(x, y) = 3x2y, r has vertices (−5, 0), (−5, 3), (5, 3), (5, 0). g

We want to find the average value of f(x, y) on the given region.

The average value of f(x, y) in the given rectangle is 25.

So we want to find the average value of f(x, y) = 3*x^2*y on the rectangle with vertices (−5, 0), (−5, 3), (5, 3), (5, 0).

This will be given by the quotient between the integral of the function in that region and the area of the region, so we have:

Where we just solved the integral in the denominator, which just gives the area of the rectangle.

Now we need to solve the integral in the numerator, we can do it one variable at the time.

This means that the average value of f(x, y) in the given rectangle is 25.

If you want to learn more, you can read:

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step-by-step explanation:

x =±sqrt(1/3)

step-by-step explanation:

x+4/6x=1/x

multiply each side by 6x to get rid of the fractions

6x*(x+4/6x)=1/x*6x

distribute

6x^2 +4 = 6

subtract 4 from each side

6x^2 +4-4 = 6-4

6x^2 =2

divide by 6

6x^2/6 = 2/6

x^2 = 1/3

take the square root of each side

sqrt(x^2) = ±sqrt(1/3)

x =±sqrt(1/3)