Science:Math Exam Resources/Courses/MATH200/April 2012/Question 06 (a)/Solution 1

If we examine the region of integration by drawing a picture...

The domain of integration is enclosed within the curves $\displaystyle x=y-2$ (black), ${\color {blue}x=-{\sqrt {y}}}$ (blue), and ${\color {red}x={\sqrt {y}}}$ (red)

... we see that if we integrate with respect to y first, that we can specify the lower bound of integration as $y=x^{2}$ and the upper bound as $y=x+2$. Looking at where the lines meet, we can see that the bounds of integration for x is x = -1 to x = 2. (To find these intersection points solve the equation $x^{2}=x+2$.)

From these observations, we can write I as follows: ${\color {blue}I=\int _{-1}^{2}\int _{x^{2}}^{x+2}f(x,y)\,{\textrm {d}}y{\textrm {d}}x}$