Question

Reversing the order of integration: Sketch the region determined by the limits of the given iterated integrals, and then evaluate the integrals by reversing the order of integration.

${\int }_0^4{\int }_y^4{e}^{\frac{y}{x}}dxdy$

Short answer

${\int }_0^4{\int }_y^4{e}^{\frac{y}{x}}dxdy=8(e-1)$

Step-by-step solution

Draw the region

The region determined by the limits of the given iterated integral is shown below,

 Reversing the order of integration 

${\int }_0^4{\int }_y^4{e}^{\frac{y}{x}}dxdy\Rightarrow {\int }_0^4{\int }_0^x{e}^{\frac{y}{x}}dxdy$

Evaluate the integral

$$\begin{gathered} I={\int }_0^4\left[{\int }_0^x{e}^{\frac{y}{x}}dy\right]dx \\ I={\int }_0^4{\left[\frac{e^{{\displaystyle \frac{y}{x}}}}{1/x}\right]}_0^{x}dx \\ I={\int }_0^{4}x\left[e-1\right]dx \\ I=(e-1){\left[\frac{x^2}{2}\right]}_0^4 \\ I=8(e-1) \end{gathered}$$