Find the average value of the given function
F(x, y)

Over the plane region R.
; R is the triangle with vertices (0, 0), (2, 0) and (2, 2).

Question

Find the average value of the given function ​
F(x, y)
​
Over the plane region R.
​  ; R is the triangle with vertices (0, 0), (2, 0) and (2, 2).
​

Quizplus · Accepted Answer

To find the average value of the function over the given region, we need to calculate the double integral of the function over the region divided by the area of the region.

The region R is a triangle with vertices (0,0), (2,0), and (2,2).

First, we need to find the limits of integration.

Since the bottom of the triangle is on the x-axis, we can integrate from x = 0 to x = 2.

The limits for y depend on which side of the triangle we are on. If we draw a vertical line from (0,0) to (2,2), we see that on the left side of the triangle, y goes from y = 0 to y = x, while on the right side, y goes from y = 0 to y = 2-x.

So we can set up the integral as:

integral from 0 to 2 of the integral from 0 to x of F(x,y) dy dx   (for the left side of the triangle) 
+
integral from 0 to 2 of the integral from 0 to 2-x of F(x,y) dy dx   (for the right side of the triangle)

To find the average, we need to divide the double integral by the area of the region, which is (1/2) base times height = (1/2) * 2 * 2 = 2.

So the average is:

(1/2) * ( integral from 0 to 2 of the integral from 0 to x of F(x,y) dy dx   +
integral from 0 to 2 of the integral from 0 to 2-x of F(x,y) dy dx ) / 2

Using the given function F(x, y) = e^(x+y), we can evaluate the double integral as follows:

integral from 0 to 2 of the integral from 0 to x of e^(x+y) dy dx  +  
integral from 0 to 2 of the integral from 0 to 2-x of e^(x+y) dy dx

= integral from 0 to 2 of [ e^(x+y) ] evaluated from y=0 to y=x dx +
integral from 0 to 2 of [ e^(x+y) ] evaluated from y=0 to y=2-x dx

= integral from 0 to 2 of ( e^(2x) - e^x ) dx + integral from 0 to 2 of ( e^(2-x) - e^x ) dx

= ( e^4 - 2e^2 + 1 )/2 + ( e^2 - e )/2

= ( e^4 + e^2 - 2e + 1 )/2

So the average value of F(x,y) over R is:

(1/2) * ( e^4 + e^2 - 2e + 1 )/2

= ( e^4 + e^2 - 2e + 1 )/4

= 11.7 (to one decimal place)

Therefore, the best choice is B.

Find the Average Value of the Given Function ​

Find the Average Value of the Given Function