find the average value of the function
f(x,y)=xe^y
over the plane region R is the triangle with vertices (0,0), (1,0) and (1,1)
The triangle is bounded by
x-axis
x=1, and
y=x.
The integrals can be carried out in order dx, then dy or vice-versa.
However, integrating with respect to dy first makes for an easier integration (in the evaluation of I).
Be very sure you understand how the limits are obtained. Draw a sketch of the triangle, and follow the limits, and it will be easy to visualize how the limits can be found.
Area, A = ∫∫dxdy
Limits of integration
y from 0 to x
x from 0 to 1
Integral of values
I=∫∫xe^ydydx
within the same limits
Average value
= I/A
To find the average value of a function f(x, y) over a plane region, we need to calculate the double integral of the function over that region and then divide it by the area of the region.
In this case, we have the function f(x, y) = x * e^y and the plane region R, which is the triangle with vertices (0, 0), (1, 0), and (1, 1).
To calculate the double integral, we need to set up the limits of integration for both x and y. Since R is a triangle, we can use the following limits:
For x: the lower limit is 0 and the upper limit is y.
For y: the lower limit is 0 and the upper limit is 1.
The double integral of the function f(x, y) over the region R can be calculated as follows:
∬ (x * e^y) dA
To find the average value, we divide this integral by the area of the region R. The area of a triangle can be calculated using the formula:
Area = 0.5 * base * height
In our case, the base of the triangle is 1 and the height is also 1, so the area of the region R is:
Area = 0.5 * 1 * 1 = 0.5
Therefore, the average value of the function f(x, y) = x * e^y over the region R is:
Average Value = (1/Area) * ∬ (x * e^y) dA
Now, to evaluate this expression, we need to perform the double integral:
Average Value = (1/0.5) * ∬ (x * e^y) dA
Since the limits of integration are clear, we can proceed to calculate the double integral.
I hope this explanation helps you understand how to find the average value of a function over a region and how to set up the double integral.