Question

\(\int\limits_1^2 {\int\limits_0^{lnx} {f(x,y)dy} dx} \)

Short answer

We should know how to do graphical representation.

Step-by-step solution

Graphical representation:

Consider,\(\int\limits_1^2 {\int\limits_0^{\ln x} {f(x,y)dy} dx} \)

The inner limits of integration indicate that \(0 \le y \le \ln (x)\)and the outer limits of integration indicate that \(1 \le x \le 2\).since the two curves that limit \(y\)intersect at \(x = 1\)it is likely that the region of integration is determined by three curves.

\(y = 0,y = \ln (x)\)and \(x = 2\).

Graph these three curve.

Graph:

Original integral:

Let us integrate on \(x\) first find the limits of a horizontal line integrating the region if integration. This horizontal line starts at \(y = \ln (x)\) and ends at \(x = 2\). The former can be solved for\(x\) as \(x = {e^x}\).

The horizontal line can slide from \(y = 0\) to \(y = \ln (x)\).

Therefore, the original integral can be written as \(\int\limits_0^{\ln (x)} {\int\limits_{{e^y}}^2 {f(x,y)dx} dy} \).