Evaluate the improper integral.
-$$\int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } e ^ { - ( 10 x + 7 y ) } d y d x$$
A) $\frac { 1 } { 140 }$
B) $\frac { \pi } { 70 }$
C) $\frac { 1 } { 70 }$
D) $\frac { \pi } { 140 }$

Question

Quizplus · Accepted Answer

To evaluate this integral, we use Fubini's theorem to interchange the order of integration. Thus, we have:
$$\int_{0}^{\infty}\int_{0}^{\infty}e^{-(10x+7y)}dydx=\int_{0}^{\infty}\left(\int_{0}^{\infty}e^{-(10x+7y)}dx\right)dy$$
We can evaluate the inner integral as:
$$\int_{0}^{\infty}e^{-(10x+7y)}dx=\left[\frac{e^{-(10x+7y)}}{-10}\right]_{0}^{\infty}=\frac{e^{-7y}}{10}$$
Substituting this into the outer integral, we get:
$$\int_{0}^{\infty}\frac{e^{-7y}}{10}dy=\left[-\frac{e^{-7y}}{70}\right]_{0}^{\infty}=\frac{1}{70}$$
Therefore, the correct answer is (C).

Evaluate the Improper Integral ∫0∞∫0∞e−(10x+7y)dydx\int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } e ^ { - ( 10 x + 7 y ) } d y d x∫0∞​∫0∞​e−(10x+7y)dydx

Evaluate the Improper Integral $\int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } e ^ { - ( 10 x + 7 y ) } d y d x$