# Integrate the function f(x, y) = ye^{yx} over the region D bounded by the lines x = 1/2, y = 1/3 ...

## Question:

Integrate the function {eq}f(x, y) = ye^{yx} {/eq} over the region {eq}D {/eq} bounded by the lines {eq}\displaystyle x = \frac 12, y = \frac 13 {/eq} and the graph of {eq}\displaystyle y = \frac 1x. {/eq}

## Double Integral:

A double integral can be integrated with two possible ways:

1. Integrate with respect to two functions of y with respect to x, then integrate along with the interval of x.

2. Integrate with respect to two functions of x with respect to y, then integrate along with the interval of y.

For this problem, integrate first from x = 1/2 to x = 1/y, then integrate from y = 1/3 to y = 2.

{eq}\displaystyle I = \int_{1/3}^2 \int_{1/2}^{1/y} ye^{yx} \ dx \ dy \\ \displaystyle I = \int_{1/3}^2 \int (e^{yx})_{1/2}^{1/y} \ dy \\ \displaystyle I = \int_{1/3}^2 \int e-e^{\frac{1}{2}y} \ dy \\ \displaystyle I = \left(ey - 2e^{\frac{1}{2}y} \right)_{1/3}^2 \\ \displaystyle I = \frac{5}{3} e - 2(e-e^{1/6}) \\ \displaystyle I = \frac{1}{3} e + e^{1/6} \\ {/eq} 