Evaluate the following iterated integral. $\int _ { 5 } ^ { 8 } \int _ { 1 } ^ { \sqrt { x } } 2 y e ^ { - x } d y d x$
A) $5 e ^ { 5 } - 8 e ^ { 8 }$
B) $5 e ^ { - 5 } - 8 e ^ { - 8 }$
C) $5 e ^ { - \sqrt { 5 } } - 8 e ^ { - \sqrt { 8 } }$
D) $5 e ^ { - 5 } + 8 e ^ { - 8 }$
E) $8 e ^ { - 8 } - 5 e ^ { - 5 }$

Question

Quizplus · Accepted Answer

The integral with respect to $ y $ evaluates to $ y^2 e^{-x} $ from $ 1 $ to $ \sqrt{x} $, resulting in $ (x - 1)e^{-x} $. Integrating with respect to $ x $ from $ 5 $ to $ 8 $ gives $ 5 e^{-5} - 8 e^{-8} $.

Evaluate the Following Iterated Integral ∫58∫1x2ye−xdydx\int _ { 5 } ^ { 8 } \int _ { 1 } ^ { \sqrt { x } } 2 y e ^ { - x } d y d x∫58​∫1x​​2ye−xdydx

Evaluate the Following Iterated Integral $\int _ { 5 } ^ { 8 } \int _ { 1 } ^ { \sqrt { x } } 2 y e ^ { - x } d y d x$