To find the average value of the function \( f(x, y) = x e^y \) over a specific triangular region, you typically use the following formula for the average value of a function over a region \( R \):
\[ \text{Average value} = \frac{1}{A(R)} \iint_R f(x, y) , dA \]
where \( A(R) \) is the area of the region \( R \), and \( \iint_R f(x, y) , dA \) is the double integral of the function over the region.
Step 1: Identify the triangular region
To proceed, it's important to define the vertices of the triangle for which you want to evaluate the average. Since we don't have the specifics about the triangle from the Questions LLC website, let's assume the triangle has vertices at \( (0, 0) \), \( (a, 0) \), and \( (0, b) \).
Step 2: Calculate the area of the triangle
The area \( A \) of a triangle with base \( a \) and height \( b \) is given by:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times a \times b \]
Step 3: Set up the double integral
The double integral over the triangular region can be accomplished using appropriate limits. In this case, we can integrate x from 0 to a and for each x, y from 0 to the line connecting the points \((a, 0)\) and \((0, b)\). The equation of the line is \( y = -\frac{b}{a}x + b \).
The double integral can be set up as:
\[ \iint_R f(x, y) , dA = \int_0^a \int_0^{-\frac{b}{a}x + b} x e^y , dy , dx \]
Step 4: Evaluate the inner integral
First, evaluate the inner integral:
\[ \int_0^{-\frac{b}{a}x + b} x e^y , dy \]
Since \( x \) is treated as a constant with respect to \( y \):
\[ = x \left[ e^y \right]_0^{-\frac{b}{a}x + b} = x \left( e^{-\frac{b}{a}x + b} - 1 \right) \]
Step 5: Evaluate the outer integral
Now we integrate with respect to \( x \):
\[ \int_0^a x \left( e^{-\frac{b}{a}x + b} - 1 \right) , dx \]
This requires splitting the integral into two parts, one involving the exponential and the other as a simple polynomial.
Evaluating these integrals will give the total value of the double integral.
Step 6: Compute the average value
Finally, once you have the double integral, compute the average using the area \( A \) calculated in step 2:
\[ \text{Average value} = \frac{1}{A} \cdot \text{Double integral} \]
Once the area and the integrals have been computed, you will have the average value of \( f(x, y) = x e^y \) over the specified triangle.
Note: Exact calculations and the resulting average will depend on the specific coordinates and parameters defined for the triangle in question. If you can provide the vertices or dimensions of the triangle, I can help perform the calculations more specifically.