Evaluate the iterated integral.
$$
\int_{-1}^{1}\left(\int_{-2}^{2} e^{x}+e^{y} d y\right) d x=
$$

Choose 1 answer:
(A) $2 e^{2}+4 e-4 e^{-1}-2 e^{-2}$
(B) $e^{2}-4 e-4 e^{-1}+e^{-2}$
(C) $4 e^{2}+4 e^{-2}$
(D) $2 e^{2}-2 e^{-2}$

Question

Evaluate the iterated integral.
$$
\int_{-1}^{1}\left(\int_{-2}^{2} e^{x}+e^{y} d y\right) d x=
$$

Choose 1 answer:
(A) $2 e^{2}+4 e-4 e^{-1}-2 e^{-2}$
(B) $e^{2}-4 e-4 e^{-1}+e^{-2}$
(C) $4 e^{2}+4 e^{-2}$
(D) $2 e^{2}-2 e^{-2}$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Evaluate the inner integral with respect to $y$: $\int_{-2}^{2} e^{x} + e^{y} \, dy$ ‖ ‖ step 2 ⋮ Since $e^{x}$ is a constant with respect to $y$, the integral becomes $e^{x} \int_{-2}^{2} dy + \int_{-2}^{2} e^{y} \, dy$ ‖ ‖ step 3 ⋮ Compute the integrals: $e^{x} \cdot (y \big|_{-2}^{2}) + (e^{y} \big|_{-2}^{2})$ ‖ ‖ step 4 ⋮ Substitute the limits of integration: $e^{x} \cdot (2 - (-2)) + (e^{2} - e^{-2})$ ‖ ‖ step 5 ⋮ Simplify the expression: $4e^{x} + (e^{2} - e^{-2})$ ‖ ‖ step 6 ⋮ Evaluate the outer integral with respect to $x$: $\int_{-1}^{1} (4e^{x} + e^{2} - e^{-2}) \, dx$ ‖ ‖ step 7 ⋮ Split the integral: $4 \int_{-1}^{1} e^{x} \, dx + \int_{-1}^{1} (e^{2} - e^{-2}) \, dx$ ‖ ‖ step 8 ⋮ Since $e^{2} - e^{-2}$ is constant with respect to $x$, the second integral becomes $(e^{2} - e^{-2}) \cdot (x \big|_{-1}^{1})$ ‖ ‖ step 9 ⋮ Compute the integrals: $4(e^{x} \big|_{-1}^{1}) + (e^{2} - e^{-2}) \cdot (1 - (-1))$ ‖ ‖ step 10 ⋮ Substitute the limits of integration: $4(e^{1} - e^{-1}) + (e^{2} - e^{-2}) \cdot 2$ ‖ ‖ step 11 ⋮ Simplify the expression: $4(e - e^{-1}) + 2(e^{2} - e^{-2})$ ‖ ‖ step 12 ⋮ Factor out common terms: $4e - 4e^{-1} + 2e^{2} - 2e^{-2}$ ‖ ‖ step 13 ⋮ Combine like terms: $2e^{2} + 4e - 4e^{-1} - 2e^{-2}$ ‖ ***A*** ∻Key Concept∻ ⚹Iterated Integration⚹ ∻Explanation∻ ⚹When evaluating an iterated integral, we integrate with respect to one variable at a time while treating the other variable as a constant. After integrating with respect to the inner variable, we substitute the limits of integration for that variable, and then integrate the resulting expression with respect to the outer variable.⚹◊What is the result of evaluating the given iterated integral involving exponential functions and specified limits of integration?⍭ Generate me a similar question◊

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