Find the average value of the given function f(x, y)
Over the plane region R. ; R is the region bounded by the graph of y = 2x and y = 0 from x = 1 to x = 3.

Question

Find the average value of the given function f(x, y)
Over the plane region R.  ; R is the region bounded by the graph of y = 2x and y = 0 from x = 1 to x = 3.

Quizplus · Accepted Answer

The average value of a function f(x,y) over a plane region R is given by:

$\frac{1}{Area(R)} \iint\limits_{R} f(x,y) dA$

In this case, the region R is bounded by the graphs y = 2x and y = 0 from x = 1 to x = 3. So, the area of R is:

$Area(R) = \int_{1}^{3} \int_{0}^{2x} dy dx = \int_{1}^{3} 2x dx = 4$

Now, we need to find:

$\frac{1}{4} \iint\limits_{R} f(x,y) dA$

The function f(x,y) is given by:

$f(x,y) = x^2y$

So, we have:

$\frac{1}{4} \iint\limits_{R} x^2y dA = \frac{1}{4} \int_{1}^{3} \int_{0}^{2x} x^2y dy dx$

$= \frac{1}{4} \int_{1}^{3} x^3 dx = \frac{80}{4} = 20$

Therefore, the average value of f(x,y) over R is 20.

Find the Average Value of the Given Function F(x, Y)