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Problem 254

# Find $$\iint_{D^{*}} \exp [(x-y) /(x+y)] d x$$ dy where $$D^{*}$$ is the region bounded by the lines $$\mathrm{x}=0, \mathrm{y}=0$$, and $$\mathrm{x}+\mathrm{y}=1$$.

Expert verified
The short answer is: $$\iint_{D^{*}} \exp\left[\frac{x-y}{x+y}\right] dx dy = \frac{1}{2}\exp[1] - \frac{1}{4}\exp[-1]$$.
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## Step 1: Set up the limits of integration for x

Since x = 0 and x + y = 1, we can find the limits of integration for x in terms of y as: $$0 \le x \le 1 - y$$

## Step 2: Set up the limits of integration for y

As y = 0 and y + x = 1 (with x = 0), the limits of integration for y are: $$0 \le y \le 1$$ Now, we have the limits of integration for both x and y. We can now integrate the given function with respect to both x and y.

## Step 3: Integrate with respect to x

Let's first integrate the function with respect to x. The integral becomes: $$\int_{0}^{1}\int_{0}^{1-y}\exp\left[\frac{x-y}{x+y}\right]dx dy$$ Perform the integration with respect to x: $$\int_{0}^{1}\left[\frac{x+y}{2}\exp\left[\frac{x-y}{x+y}\right]\right]_{x = 0}^{x = 1 - y} dy$$

## Step 4: Evaluate the integral and simplify

Now we need to evaluate the integral with respect to x: $$\int_{0}^{1}\left(\frac{1}{2}\exp[1] - \frac{y}{2}\exp[-1]\right) dy$$ Now, we proceed to integrate with respect to y.

## Step 5: Integrate with respect to y

Integrate the given expression with respect to y: $$\left[\frac{1}{2}\exp[1]y - \frac{y^2}{4}\exp[-1]\right]_{y = 0}^{y = 1}$$

## Step 6: Evaluate the integral and obtain the final result

Evaluate the integral and find the final result for the double integral: $$\left[\frac{1}{2}\exp[1] - \frac{1}{4}\exp[-1]\right] - \left[0\right] = \frac{1}{2}\exp[1] - \frac{1}{4}\exp[-1]$$ So, the double integral of the given function over the domain $$D^{*}$$ is: $$\iint_{D^{*}} \exp\left[\frac{x-y}{x+y}\right] dx dy = \frac{1}{2}\exp[1] - \frac{1}{4}\exp[-1]$$

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