Problem

$\int_{3}^{5} \int_{1}^{\sqrt{x}} 2 y e^{-x} d y d x$

Answer

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Answer

The final answer is \(\boxed{-5 e^{-5} + 3 e^{-3}}\)

Steps

Step 1 :Given the double integral problem: \(\int_{3}^{5} \int_{1}^{\sqrt{x}} 2 y e^{-x} d y d x\). The inner integral is with respect to y and the outer integral is with respect to x. The function to be integrated is \(2 y e^{-x}\). The limits of the inner integral are from 1 to \(\sqrt{x}\) and the limits of the outer integral are from 3 to 5.

Step 2 :First, solve the inner integral, treating x as a constant. The function to be integrated becomes \(2 y e^{-x}\).

Step 3 :The result of the inner integral is \(x e^{-x} - e^{-x}\).

Step 4 :Next, substitute the limits of the inner integral into the result. This gives us the function to be integrated for the outer integral.

Step 5 :Then, solve the outer integral, treating the result of the inner integral as the function to be integrated.

Step 6 :The result of the outer integral is \(-5 e^{-5} + 3 e^{-3}\).

Step 7 :Finally, substitute the limits of the outer integral into the result to get the final answer.

Step 8 :The final answer is \(\boxed{-5 e^{-5} + 3 e^{-3}}\)

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