00:02
So we begin this problem by graphing the region of interest, and that is the region that is bounded on the top by the blue curve, y equals 8 minus x squared, below by the curve in green, y equals 2x, and on the left by the curve in red, which is x equals 1.
00:24
And i've outlined, or i've highlighted that region here in black.
00:29
We do need to know at some point in the future clearly what the point of intersection will be so that we may complete the remainder of the problem.
00:40
So we're interested in where 2x is equal to 8 minus x squared.
00:45
So we're looking at x squared plus 2x minus 8 equals 0.
00:50
And this occurs when we can factor and solve at x equals negative 4 and positive 2.
01:00
This is clearly the point 2 comma 4 for our point of intersection.
01:06
Part b wants us to find the area of this curve, and probably the easiest thing to do is imagine a representative rectangle here shown in green that uses dx, where the top curve is 8 minus x squared, the bottom curve is 2x.
01:24
So for part b of this problem, our area between the two curves will be the integral from 1 .5 .5 .2 .2 .5.
01:31
To 2.
01:32
That is the boundary on the left of x equals 1 to the point of intersection on the right x equals 2...