The area of the region enclosed between the two curves given...

The area of the region enclosed between the two curves given by C1 : y = x^2 - 2x + 21 and C2 : y = 2x^2 - x + 1 is evaluated via ??? (f(x) - g(x)) dx = (ax^3 + bx^2 + cx) |?? where ?= ?= a= b= c= Evaluating this integral gives the Area: Note: You can earn partial credit on this problem.

Transcript

00:02 So in this question, they say that the area of the region enclosed between the two curves, given by c1, which is y equals x squared minus 2x plus 21, and c2, which is y equals 2x squared minus x plus 1, is evaluated by some definite integral, and we need to figure out the value of all of these constants, as well as that area.

00:31 So let's start by making the sketch of our region.

00:35 So let's look at the curve, y equals x squared minus 2x plus 21, as well as the curve, 2x squared, minus x plus 1.

00:52 So let's go ahead, and if we were to graph this in a standard viewing window, i might not get the best picture.

00:59 So let me change my window.

01:02 Let me make my x max larger.

01:05 Let's go up to 30 and see if that's enough.

01:08 Not quite.

01:09 I'm going to go up a little bit higher.

01:12 So i'm going to go up to maybe 60s.

01:15 Let me change this window yet again.

01:17 Make my y max 60.

01:21 And now i see the region in question.

01:24 Now i need to figure out where these curves intersect.

01:29 So where do these guys? intersect.

01:34 So my first intersection is at x equals negative 5 and my second intersection is going to be at positive 4.

01:50 Now if you wanted to see that algebraically, i would set these equal to each other to see where they intersect.

01:58 If you set x squared minus 2x plus 21 equal to 2x minus 2x plus 21 equal to 2x squared minus x plus 1.

02:08 I'm going to set this to 0.

02:10 I subtract the x squared.

02:15 I'm going to add over the 2x, and i subtract the 21.

02:22 What's here on the right now factors.

02:25 It factors as the quantity of x plus 5 times the quantity of x minus 4, giving us x equals negative 5 and positive 4.

02:37 And so we are integrating on the interval, from negative 5 to positive 4.

02:44 And i'm integrating my top function minus my bottom function.

02:49 My top function this time is the blue curve, the quadratic, that has a coefficient of 1 as its leading coefficient.

03:00 So, x squared minus 2x plus 21.

03:05 And from this, i am subtracting the 2x squared minus x plus 1.

03:11 All of this dx...

Question

The area of the region enclosed between the two curves given by C1 : y = x^2 - 2x + 21 and C2 : y = 2x^2 - x + 1 is evaluated via ??? (f(x) - g(x)) dx = (ax^3 + bx^2 + cx) |?? where ?= ?= a= b= c= Evaluating this integral gives the Area: Note: You can earn partial credit on this problem.

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