By using the formula: A = [x - (x2 - 2) ]dx - 1 = (x - x2 - 2) dx 1 We may now get the integrals: x2 2 3 + 2x We can now substitute with the intervals: 22 23 (-1)2 + 2(2) (-1)3 2 3 + 2(-1) 2 3 = (2-3+4) -(2+3-2) Giving us the answer: A REFLECTION: With the answer that has been given, we can now conclude that the plane region is 9/2 or 4.5, with this,PLANE REGIONS: 1. Ms. Batumbakal jr. Is scratching her head over her test paper for the upcoming mathematics quiz bee, the question was "Find the plane region bounded the curves y = x2 - 2 and y = x We must first get the points of intersection between the two curves y = x2 - 2 And we can see that y = x meaning we can replace y with x like so: x =x2 - 2 We can expand upon this by factoring 0 = x2 -x -2 0 = (x - 2) (x + 1) We now have two possible equations x - 2 =0 |x+1=0 x =2 |x=-1 With y = x in mind, x = 2 Is also the same with y = 2 The same can also be said with x = -1 and y = -1 Both giving us (2,2) and (-1, -1) -2 Looking at the graph, we can see that they both intersect at (2,2) and (-1, -1). We may now proceed with using the formula for the Area of Region: A = [ If () -9(x)]dx We must first find the functions for the formula To getf (x), it is the shaded region bounded above, and it is y = x The same goes for g (x), this time it is the shaded region bounded below, and it is y = x2 - 2 We now get, f(x) = x and g(x) =x2 -2 Then, we now get the two intervals The first one is on x = -1 Then the other one is in x = 2 Giving us: a=-1 and b = 2