Evaluate the integral.
-$$\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { y } e ^ { x + y } d x d y$$
A) $\frac { 1 } { \mathrm { e } } \left( \mathrm { e } ^ { 2 } - \mathrm { e } \right) ^ { 2 }$
B) $\frac { 1 } { 3 } ( \mathrm { e } - 1 ) ^ { 2 }$
C) $\frac { 1 } { 2 } ( \mathrm { e } - 1 ) ^ { 2 }$
D) $\frac { 1 } { 2 } \left( e ^ { 2 } - e \right) ^ { 2 }$

Question

Quizplus · Accepted Answer

By reversing the order of integration, we get
$$\int_{0}^{1}\int_{x}^{1}e^{x+y}\,dy\,dx=\int_{0}^{1} e^{x} \int_{x}^{1} e^{y}\,dy\,dx=\int_{0}^{1} e^{x} \left(e^{1}-e^{x}\right)\,dx=\frac{1}{2}(e-1)^2.$$ So, the answer is C.

Evaluate the Integral ∫01∫0yex+ydxdy\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { y } e ^ { x + y } d x d y∫01​∫0y​ex+ydxdy

Evaluate the Integral $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { y } e ^ { x + y } d x d y$