## f(x,y)=2exy

We need to find the average value of f(x,y)=2exy over the given rectangle R: {(x,y) | 0≤x≤12, 0≤y≤1}. We can do this by finding the integral of f(x,y) over R and divide it by the area of R.

### Set up the integral

We need to find the average value of f(x,y)=2exy over the given rectangle R. To do this, we’ll set up an integral:

$$\int_0^1\int_{12}^{2e}\frac{f(x,y)}{(1)(2e)}dydx$$

The integrand is the function f(x,y) divided by the rectangle’s width (1) multiplied by its height (2e). Now we can solve the integral:

$$\int_0^1\int_{12}^{2e}\frac{f(x,y)}{(1)(2e)}dydx=\int_0^1\left(\int_{12}^{2e}\frac{2exy}{2e}dy\right)dx$$

$$=\int_0^12exydx=24ex-24e^2x+12ex^2 \bigg|_{12}^{2e}$$

$$=24e^{12}-24e^{14}+288e-216e^2+144 e^{13}-288 e^{15} \bigg|_0^1$$

### Find the average value of f over the given rectangle

To find the average value of f(x,y) over the given rectangle, we need to first calculate the value of f(x,y) at each point in the rectangle. We then take the sum of all these values and divide by the number of points in the rectangle.

The value of f(x,y) at each point in the rectangle is:

f(x,y)=2exy – (find average value of f over given rectangle)

Therefore, the average value of f(x,y) over the given rectangle is:

AverageValueOfF=(2exy-(0)+2exy-(1)+2exy-(2)+…+2exy-(12))/144

AverageValueOfF=(2exy-12)/144

=24e/144

=e/6

## f(x,y)=exy

f(x,y)=exy is a function that is defined over the given rectangle r 0 12 0 1. To find the average value of this function over the given rectangle, we will need to take a integral of the function.

### Set up the integral

We need to find the average value of f(x,y)=exy over the given rectangle R with vertices at (0,0), (12,0), (12,1), and (0,1). To do this, we’ll set up an integral:

average value of f

= 1/Area of R * integral from x=0 to x=12 integral from y=0 to y=1 f(x,y) dy dx

Now we can plug in our function for f(x,y):

average value of f

= 1/Area of R * integral from x=0 to x=12 integral from y=0 to y=1 exy dy dx

To find the Area of R, we can simply multiply the length and width:

Area of R = length * width

= 12 * 1

= 12

### Find the average value of f over the given rectangle

Calculate the average value of the function over the given rectangle by dividing the integral of the function over the rectangle by the area of the rectangle. In this case, f(x,y)=exy and the given rectangle is fx y 2ey x ey r 0 12 0 1.

average value = integral of f(x,y) over rectangle / area of rectangle

= (∫f(x,y)dx*∫f(x,y)dy)/((b-a)*(d-c))

= (∫0 to 1 ∫0 to 12 exydx*dy)/((12-0)*(1-0))

= ((∫0 to 12 exydy)*(∫0 to 1 exydx))/((12-0)*(1-0))

= ((∫0 to 12 ey*dy)*(∫0 to 1 exydx))/12

= (12*ey – 0)/12

= ey

## f(x,y)=xey

### Set up the integral

We need to find the average value of f(x,y)=xey over the given rectangle. This is equivalent to finding the integral of f(x,y) over the rectangle and then dividing by the area of the rectangle.

The area of the rectangle is (12-0)*(1-0)=12.

So, the average value of f(x,y) over the rectangle is:

int_0^12 int_0^1 f(x,y) dy dx/12=

=int_0^12 xey dy/12=

Now we need to find the limits of integration for x and y. For y, we have 0<=y<=1 since that is one side of the rectangle. For x, we have 0<=x<=12 since that is the other side of the rectangle. So our final answer is:

=int_0^12 int_0^1 xey dy dx/12=

### Find the average value of f over the given rectangle

To find the average value of a function over a given rectangle, we need to first calculate the function’s values at the four corners of the rectangle, (x, y), (x + h, y), (x, y + k), and (x + h, y + k). We then take the mean of these four values.

In this case, the given rectangle is f(x,y) = xey – 2ey x + ey r 0 12 0 1 . The four corners of the rectangle are:

(0,0): f(0,0) = 00 – 20 + 01 = 01

(12,0): f(12,0) = 126 – 2412 + 012 = 1212

(0,1): f(0,1) = 00 – 20 + 11 = 11

(12,1): f(12.1) = 126 – 2412 + 112 = 12112

The average value of f over the given rectangle is then (01+1212+11+12112)/4 = 2.75.