Answer
$$2\ln 2$$
Work Step by Step
$$\eqalign{
& \int_2^4 {\int_2^4 {\frac{1}{y}dx} dy} \cr
& \int_2^4 {\left[ {\int_2^4 {\frac{1}{y}dx} } \right]} dy \cr
& {\text{solve the inner integral}} \cr
& \int_2^4 {\frac{1}{y}dx} = \frac{1}{y}\int_2^4 {dx} \cr
& = \frac{1}{y}\left[ x \right]_2^4 \cr
& {\text{evaluate limits}} \cr
& = \frac{1}{y}\left( {4 - 2} \right) \cr
& = \frac{2}{y} \cr
& {\text{then}} \cr
& \int_2^4 {\int_2^4 {\frac{1}{y}dx} dy} = \int_2^4 {\frac{2}{y}} dy \cr
& {\text{integrating}} \cr
& = 2\left[ {\ln \left| y \right|} \right]_2^4 \cr
& evaluating \cr
& = 2\left( {\ln 4 - \ln 2} \right) \cr
& = 2\left( {\ln 4 - \ln 2} \right) \cr
& = 2\ln 2 \cr} $$
