We need to evaluate the double integral:

$\int_{0}^{1} \int_{y}^{1} x e^x \, dx \, dy$

Step 1: Inner integral

First, evaluate the inner integral:

$I = \int_{y}^{1} x e^x \, dx$

For this, use integration by parts. Let:

• $u = x$ (so $du = dx$)
• $dv = e^x dx$ (so $v = e^x$)

Using the integration by parts formula: $\int u \, dv = uv - \int v \, du$ we get: $\int x e^x \, dx = x e^x - \int e^x \, dx = x e^x - e^x + C$

Thus, the inner integral becomes: $I = \left[ x e^x - e^x \right]_{y}^{1}$

Substitute the limits of integration ($x = y$ to $x = 1$): $I = \left( 1 e^1 - e^1 \right) - \left( y e^y - e^y \right)$ $I = \left( e - e \right) - \left( y e^y - e^y \right)$ $I = 0 - (y e^y - e^y)$ $I = e^y - y e^y$

Step 2: Outer integral

Now, evaluate the outer integral: $\int_{0}^{1} \left( e^y - y e^y \right) \, dy$

Split this into two separate integrals: $\int_{0}^{1} \left( e^y - y e^y \right) \, dy = \int_{0}^{1} e^y \, dy - \int_{0}^{1} y e^y \, dy$

First term:

$\int_{0}^{1} e^y \, dy = \left[ e^y \right]_{0}^{1} = e^1 - e^0 = e - 1$

Second term:

Use integration by parts again for $\int_{0}^{1} y e^y \, dy$. Let:

• $u = y$ (so $du = dy$)
• $dv = e^y dy$ (so $v = e^y$)

Then: $\int y e^y \, dy = y e^y - \int e^y \, dy$ $\int y e^y \, dy = y e^y - e^y + C$

Now evaluate from $y = 0$ to $y = 1$: $\int_{0}^{1} y e^y \, dy = \left[ y e^y - e^y \right]_{0}^{1}$ $= \left( 1 e^1 - e^1 \right) - \left( 0 e^0 - e^0 \right)$ $= \left( e - e \right) - \left( 0 - 1 \right)$ $= 0 - (-1) = 1$

Final result:

Substitute the results back into the expression: $\int_{0}^{1} \left( e^y - y e^y \right) \, dy = (e - 1) - 1$ $= e - 2$

Thus, the value of the double integral is: $\boxed{e - 2}$

Would you like further clarification or steps explained in more detail?
Here are 5 related questions to explore:

1. How does integration by parts work, and why is it useful here?
2. Can this problem be solved by reversing the order of integration?
3. What are the geometric implications of this integral?
4. How do exponentials like $e^x$ behave under integration?
5. Can this integral be approximated numerically for different limits?

Tip: For problems involving $x e^x$, remember that integration by parts is the most effective strategy!