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Determine the average value of the function $(x, y)$ over the region $R$.$f(x, y)=3 x^{2} y^{2}, R$ is the region in Exercise 16.

$$256 / 3$$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 6

Double Integrals

Partial Derivatives

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In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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Compute the average value …

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In this question, we are given the function f x, y, equal to the 4 minus x, minus y and the region or equal to x y, and we have the x is between the zo and 2 and y is between the 2 point. Therefore, we can get the area of the region equal to 2 times 2 and equal to 4 point now. We need to find the double integral of the function f x. Now we can put a d x and the 1 here x came from 0 to 2 and i goes from 0 to 2 as well. We get equal to. We keep the altar. For now that we have d y for the inner we have, it will be, the d x so x will be. The variable y will be the constant delta derivative equal to the 4 x minus x square, up 2 minus x y evaluated and 32 that we get equal to 32 point we put 2 and 7 to 8 minus. Here we have the 2 square, minus 2 minus 2 y and then to equal to it: 26 minus 2 y d y and at erith 6 equal to the 6 y minus y square evaluated 0 to 2, and we get equal to the 12 minus the 2 Square equal to the 4 so at equal to 8 and therefore the average value equal to the double integral over the area. And i go to 8 out of 4 and equal to 2.

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