We are tasked with evaluating the double integral:

$\int_{x=0}^1 \int_{y=0}^{2-2x} e^x \, dy \, dx.$

Step-by-Step Solution:

1. Rewrite the integral with the correct order: $\int_{x=0}^1 \int_{y=0}^{2-2x} e^x \, dy \, dx.$

   Notice that $e^x$ is independent of $y$, so we can treat it as a constant while integrating with respect to $y$.

2. Evaluate the inner integral: The inner integral is: $\int_{y=0}^{2-2x} e^x \, dy.$ Since $e^x$ is constant with respect to $y$, the integral simplifies to: $e^x \int_{y=0}^{2-2x} 1 \, dy = e^x \big[y\big]_{y=0}^{2-2x}.$ Compute the bounds: $e^x \big[(2-2x) - 0\big] = e^x (2 - 2x).$

   Thus, the inner integral evaluates to: $\int_{y=0}^{2-2x} e^x \, dy = 2e^x - 2xe^x.$

3. Set up the outer integral: Substitute the result of the inner integral into the outer integral: $\int_{x=0}^1 (2e^x - 2xe^x) \, dx.$

4. Separate the integral: Break the integral into two parts: $\int_{x=0}^1 (2e^x - 2xe^x) \, dx = 2 \int_{x=0}^1 e^x \, dx - 2 \int_{x=0}^1 xe^x \, dx.$

5. Evaluate each part:

   • First term: $2 \int_{x=0}^1 e^x \, dx = 2 \big[e^x\big]_{x=0}^1 = 2 \big(e^1 - e^0\big) = 2(e - 1).$

   • Second term: To evaluate $\int_{x=0}^1 xe^x \, dx$, we use integration by parts: $\int xe^x \, dx = u \, dv \text{ where } u = x \text{ and } dv = e^x dx.$ Compute:

     • $u = x \implies du = dx$,
     • $dv = e^x dx \implies v = e^x$.

     Then: $\int xe^x \, dx = uv - \int v \, du = xe^x - \int e^x \, dx = xe^x - e^x + C.$ Applying the limits $x = 0$ to $x = 1$: [ \big[xe^x - e^x\big]_{x=0}^1 = \big[(1)e^1 - e^1\big] - \big[(0)e^0 - e^0\big] = (e - e) - (0 - 1) = -1. ]

     Thus: $\int_{x=0}^1 xe^x \, dx = -1.$

6. Combine the results: Substitute back into the equation: $2 \int_{x=0}^1 e^x \, dx - 2 \int_{x=0}^1 xe^x \, dx = 2(e - 1) - 2(-1).$ Simplify: $2(e - 1) + 2 = 2e - 2 + 2 = 2e.$

Final Answer:

$\int_{x=0}^1 \int_{y=0}^{2-2x} e^x \, dy \, dx = 2e.$

Would you like a detailed explanation for any step?

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5 Related Questions:

1. How does changing the bounds of integration affect the solution?
2. What if the integrand was $e^{xy}$ instead of $e^x$?
3. How do you solve integrals involving trigonometric functions as limits?
4. Can we switch the order of integration here, and how would it change the problem?
5. How does integration by parts work in general for $x^n e^x$?

Tip:

When faced with nested integrals, evaluate the inner integral first and look for opportunities to simplify constants and dependencies.