# Thread: Exponential double integral

1. ## Exponential double integral

Instructions state to find the average value of $\displaystyle f(x,y)$ over the region $\displaystyle R$.

$\displaystyle f(x,y) = e^{x+y}$ with the region R being a triangle with vertices at (0,0), (0,1) and (1,1).

So the base formula for taking the average is:

$\displaystyle Average = \frac{1}{A} \int_R \int f(x,y) dA$

Drawing the graph, I find it is a triangle anchored at the origin. I'm approaching it from the dy position so I have points from $\displaystyle y=0$ to $\displaystyle y=2$ and it's entering the region in question at $\displaystyle x=0$ and exiting the region at $\displaystyle x=y$.

My double integral is as follows:

$\displaystyle Average = \frac{1}{y} \int_0^1 \int_0^y (e^{x+y}) dxdy$

Here's where I run into trouble because I'm not sure how to use a U sub to my advantage on this problem. If I do $\displaystyle u = x+y$ so $\displaystyle du = 1 + y dx$, well I don't have a Y in the function, but obviously I do have a 1, so how would I approach this problem? Would I do $\displaystyle \frac{1}{y}du=1dx$ so I have:

$\displaystyle Average = \frac{1}{y^2} \int_0^1 (e^{y} + e^{2y}) dy$

Or would I do something different?

2. You set up correctly the double integral.

By makin' $\displaystyle u=x+y$ for the inner integral, we have $\displaystyle du=dx,$ we just treat $\displaystyle y$ as a constant, since we're takin' the derivative with respect to $\displaystyle x.$ Now, find the integration limits and evaluate the rest.

What does mean $\displaystyle A$? Area?

3. Originally Posted by emttim84
Instructions state to find the average value of $\displaystyle f(x,y)$ over the region $\displaystyle R$.

$\displaystyle f(x,y) = e^{x+y}$ with the region R being a triangle with vertices at (0,0), (0,1) and (1,1).

So the base formula for taking the average is:

$\displaystyle Average = \frac{1}{A} \int_R \int f(x,y) dA$

Drawing the graph, I find it is a triangle anchored at the origin. I'm approaching it from the dy position so I have points from $\displaystyle y=0$ to $\displaystyle y=2$ and it's entering the region in question at $\displaystyle x=0$ and exiting the region at $\displaystyle x=y$.

My double integral is as follows:

$\displaystyle Average = \frac{1}{y} \int_0^1 \int_0^y (e^{x+y}) dxdy$
What's the 1/y doing outside the integral? The quantity A is supposed to be the area of the region over which you are taking the average. Since this is a triangle, with base and height both equal to 1, its area is 1/2, so 1/A should be 2.

Originally Posted by emttim84
Here's where I run into trouble because I'm not sure how to use a U sub to my advantage on this problem. If I do $\displaystyle u = x+y$ so $\displaystyle du = 1 + y dx$, well I don't have a Y in the function, but obviously I do have a 1, so how would I approach this problem?
You don't need to make a substitution. Think of the integral as $\displaystyle 2\int_0^1\!\! \int_0^y e^{x+y}\, dxdy = 2\int_0^1\! \left(\int_0^y e^xe^y\, dx\right)dy$. As far as the inner integral is concerned, y is a constant, so you can integrate it as $\displaystyle 2\int_0^1 \!\!\left(e^y\bigl[ e^x\bigr]_0^y\right)dy = 2\int_0^1\!\!(e^{2y}-e^y)dy$. The rest I'll leave to you.

4. Wow, yeah I feel kinda embarrassed after missing the fact that $\displaystyle x+y$ with Y as the constant is a lot easier than what I must have been thinking, $\displaystyle xy$.

So with the area being 2, here's what I have:

$\displaystyle Average = 2 \int_0^1 (e^{2y} - e^y) dy$

Integrating both terms, I have:

$\displaystyle 2 \frac{1}{2} [e^{2y} - e^y]$ so the constants in front cancel out.

Evaluating them from 0 to 1 I have:

$\displaystyle Average = [(e^2 - e^1) - (e^0 - e^0)] = e^2 - e^1$

Final answer: $\displaystyle Average = e^2 - e^1$

Look good to you guys?

5. No, the value of the double integral is $\displaystyle (e-1)^2.$ Check your computations when evaluating the last integral.

6. Ok, integrating them without making algebra mistakes, I have:

$\displaystyle Average = 2 \int_0^1 (e^{2y} - e^y) dy = 2 (\frac{1}{2}e^{2y} - e^y)$ evaluated from 0 to 1 which simplifies to:

$\displaystyle Average = e^{2y} - 2e^y$ evaluated from 0 to 1 which results in:

$\displaystyle Average = [(e^{2(1)} - 2e^1) - (e^{2(0)} - 2e^0)]$

Simplifes to:

$\displaystyle Average = e^2 - 2e^1 - (-1) = e^2 - 2e^1 + 1$

And $\displaystyle e^2 - 2e^1 + 1 = (e-1)^2$