1 Example 6 Find the area A between the parabola x = y 2 and the line y = x-2. Solution The first step is to determine the intersection points of this parabola and this line x=y+2: y 2 = x = y+2, 0= y 2 -y –2 = (y+1)(y-2). Hence the intersection points are y=-1 and y=2, i.e. the points (1,-1) and (4,2). The parabola x=y 2 is one function of y but two functions of x: Therefore, it is easier to find this area by integrating with respect to y: Since D(y 3 /3 – y 2 /2 - 2y) = y 2 - y - 2,

2 An alternative approach to the preceding computation begins by sketching the region between parabola x = y 2 and the line y = x-2. From this sketch, we see that y+2  y 2 for –1  y  2. Hence x = y 2

3 Observe that we can compute the area A by integrating with respect to x. However, to do this we must divide the region into two regions by the line x=1. To the left of this line the region is bounded below by and bounded above by To the right of this line the region is bounded below by the line y=x-2 bounded above by Since D(2/3 x 3/2 ) = x 1/2, Clearly it was easier to compute the value of A by integrating with respect to y.