Write an equivalent double integral with the order of integration reversed.
-$\int _ { 1 } ^ { 2 } \int _ { 0 } ^ { \ln x } 6 x d y d x$
A) $\int _ { 0 } ^ { \ln 2 } \int _ { \mathrm { e } } ^ { 2 } 6 \mathrm { xdx } d \mathrm { y }$
B) $\int _ { 0 } ^ { \ln 2 } \int _ { 1 } ^ { 2 } 12 x d x d y$
C) $\int _ { 0 } ^ { \ln 2 } \int _ { e ^ { y } } ^ { 2 } 12 x d x d y$
D) $\int _ { 0 } ^ { \ln 2 } \int _ { 1 } ^ { 2 } 6 x d x d y$

Question

Quizplus · Accepted Answer

The original integral has limits for $x$ from 1 to 2 and for $y$ from 0 to $\ln x$. When reversing the order of integration, $y$ becomes the outer integral, and its limits are from 0 to $\ln 2$ (since $\ln x$ varies from $\ln 1 = 0$ to $\ln 2$). For $x$, given a $y$ value, $x$ must be greater than $e^y$ (since $y = \ln x$, $x = e^y$) and less than or equal to 2. Thus, the correct integral is $\int _ { 0 } ^ { \ln 2 } \int _ { e } ^ { 2 } 6 x d x d y$.

Write an Equivalent Double Integral with the Order of Integration ∫12∫0ln⁡x6xdydx\int _ { 1 } ^ { 2 } \int _ { 0 } ^ { \ln x } 6 x d y d x∫12​∫0lnx​6xdydx

Write an Equivalent Double Integral with the Order of Integration $\int _ { 1 } ^ { 2 } \int _ { 0 } ^ { \ln x } 6 x d y d x$