6. (4+3+5= 12pts.) Consider the region R enclosed by the...

6. (4+3+5= 12pts.) Consider the region R enclosed by the curves $y = x^2$ and $y - x = 2$. DO NOT USE ABSOLUTE VALUE! (a) Sketch the region and find the intersection points of the curves. Intersection points are: (-1,1) and (2, 4). (b) Write a dx-integral for the area of R. Don't calculate! $\int_{-1}^{2} (2 + x - x^2) dx$ (c) Write dy-integrals for the area of R. Don't calculate! $\int_{0}^{1} 2\sqrt{y} dy + \int_{1}^{4} (\sqrt{y} - y + 2) dy$

Transcript

00:01 In part a, we are looking at the region that is bounded by the two equations.

00:08 Y equals 2x and y equals 1 half x cubed.

00:14 So if we were to draw the first graph, when x is zero, y would be zero.

00:23 When x is 1, y would be 2, when x is 2, y would be 4, when x is 3, y would be 6.

00:31 So you are talking about a straight line.

00:36 And with the second graph, when x is zero, y would be zero.

00:41 When x is one, y would be one.

00:46 When x is two, y would be four.

00:51 And when x is three, y would be 13 and a half.

01:01 So we end up with a curve.

01:06 And we are talking about the region in between.

01:10 Here.

01:14 I'm just going to take the grid paper away at this point.

01:20 So we're talking about that purple region and we want to know the two intersection points.

01:26 So we want to know right here and right here.

01:32 So if we were to take the two equations and use substitution, since y is equal to half of x cubed, we can put that in place of y in the other equation, and we could say one half x cubed equals two x.

01:54 If we were to subtract two x from both sides, and then factor out the gcf, or greatest common factor of x, we would have one half x squared minus two equals zero, if we applied the zero product property, we get x equals zero and one half x squared minus two equals zero.

02:22 So one half of x squared is the same as two, x squared is the same as four, so x could be a plus or a minus two.

02:34 Since we are reserving our boundaries to be in the first quadrant, we're only going to focus on x being.

02:44 2.

02:45 So we know x is 0 at one intersection point and x equals 2 at the other intersection point.

02:53 And if we were substituting those two values of x into either of the two equations, we'll get their corresponding y coordinates of 0 ,0, and 24.

03:07 In part b, you want to calculate the perimeter.

03:13 So the perimeter, if i were to enlarge this picture, we're going to have a straight side and then we have a curved side.

03:27 So for me to calculate the perimeter, i want to find the length of this line...

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