Science:Math Exam Resources/Courses/MATH200/December 2013/Question 06/Solution 1

Like the hint suggests, we need to change the order of integration. First, we must begin by setting up the new bounds. We can begin by drawing the domain of integration.

$-1\leq x\leq 0$

$-2\leq y\leq 2x$

In order to change the order of integration, we must find the bounds of $x$ in terms of $y$ and the bounds of $y$ in terms of numbers.

We can observe from the picture of the domain of integration that the bounds are equivalent to

${\frac {y}{2}}\leq x\leq 0$

$-2\leq y\leq 0$

Using this, we can change the order of integration.

$\int _{-1}^{0}\int _{-2}^{2x}e^{y^{2}}dydx=\int _{-2}^{0}\int _{\frac {y}{2}}^{0}e^{y^{2}}dxdy=\int _{-2}^{0}(xe^{y^{2}}|_{x={\frac {y}{2}}}^{x=0})dy=\int _{-2}^{0}-{\frac {y}{2}}e^{y^{2}}dy\color {red}{=}$

Let $u=y^{2}$ and $du=2ydy$

$\color {red}{=}$ $-{\frac {1}{4}}\int _{4}^{0}e^{u}du={\frac {1}{4}}\int _{0}^{4}e^{u}du={\frac {1}{4}}(e^{4}-e^{0})=\color {blue}{\frac {1}{4}}(e^{4}-1)$