Sketch the region enclosed by the given curves and find its area. y=12-x^2, y=x^2-6

Name: Problem 13, Chapter 5: Calculus
Uploaded: 2020-04-19T04:04:04-08:00
Duration: 10 min 16 s
Channel: Ian Musser
Description: Sketch the region enclosed by the given curves and find its area. y=12-x^2, y=x^2-6

step 1 All right, so we're trying to answer this problem. States sketched the region, includes by the g

Sketch the region enclosed by the given curves and find its...

Key Concepts

Area Between Curves

The concept of finding the area between curves involves subtracting the lower function from the upper function and integrating this difference over the interval determined by the points of intersection. This method effectively sums up the vertical slices of area to yield the total area of the region enclosed.

Definite Integration

Definite integration is used to calculate the area between curves. Once the limits of integration are known, the area can be found by integrating the difference between the upper (or top) function and the lower (or bottom) function over these limits.

Sketching Curves

Drawing or visualizing the graphs of the given functions helps in understanding the spatial relationship between the curves. It aids in identifying which function lies above the other in the region of interest, which is necessary for correctly setting up the integral for the area calculation.

Intersection Points

Finding the points where two curves intersect is fundamental in determining the boundaries of the region they enclose. By setting the equations of the curves equal to each other, one can solve for the variable, which then provides the limits of integration or the points that define the region’s edges.

Transcript

00:01 All right, so we're trying to answer this problem.

00:04 States sketched the region, includes by the given curves and find its area.

00:09 So we're given the curves why equals 12 minus x squared, and we're giving the other pair of y equals x squared minus six.

00:23 Now, the first thing to do that we're gonna look out here, is we're gonna start with, like was 12 minus x where we're gonna look at our x squared individually.

00:33 So looking here, the first thing you notice is that this guy has got here is negative.

00:39 So i know that whatever i graph it, it's a problem that's gonna open downwards.

00:46 Now, the next thing i look at is just 12.

00:50 This 12 is gonna be our y intercept, because when access zero y equals 12 so i can just go ahead and grab that.

00:56 So that's gonna be somewhere up here.

00:58 Yeah, right in that area.

01:00 Now, the next thing we need to do is find our ex intercepts.

01:04 So we're gonna come over here and all we're going to do for ex intercepts is we're going to set our why equal to zero.

01:14 So i have zero equals 12 minus x squared, then we just use simple out about to solve for the x values.

01:22 So i'm gonna add extra both sides.

01:27 Give me 12 and then whenever i take the square root now, this is one of the key parts to remember.

01:32 You get a plus or minus square root of 12 now square of 12.

01:40 That doesn't really work out that nicely.

01:42 So just plugging into a calculator, i knew that's roughly around 3.46 and this is important just for our graphing.

01:54 So going back to our other screen here, i can say that now, around here and around here, we have our square 12 and a negative square 12.

02:16 So when that graph that i get my downward, if i can actually get it on the line down, we're open opening problem.

02:29 So we have that 1st 1 done there.

02:32 That's exactly how that looks now.

02:35 Looking at the second equation, we yet again look at the x squared.

02:39 It's a positive.

02:40 That means i know that's going to be opening up words this time.

02:43 And now look here.

02:44 I have a negative six.

02:45 So, ray, about here, down here in the negative six on the why access is gonna be.

02:52 Why? well, i intercept now.

02:54 Similar method to finding our ex intercepts.

02:59 We're just going to sit.

03:00 Why? equal to zero? so coming back over here, i'm gonna get zero equals x squared.

03:07 Last six at six.

03:09 To both sides against six equals x squared.

03:14 I swear it.

03:15 Six.

03:15 Don't forget that plus or minus.

03:19 So coming back here to graph it and square roots six is another one of those that doesn't work out very nicely.

03:30 But knowing using the calculator i know that's roughly about 2.45 so i know that it's gonna be somewhere around that midpoint.

03:43 So square it.

03:44 Six andy.

03:48 Positive square six now.

03:50 Same same deal.

03:52 Just draw your parabola through those two points and walla, you have your graph.

04:00 Now, the question in particular here is asking for all this area inside between these two.

04:10 Now we have our equations.

04:11 We have our graph.

04:13 The first thing that i need to notice from our graph is which equation is what we call bounds the top and which equation bounds the bottom.

04:24 So up here, i can see that my purple equation is the upper bound because it's the one that's on top containing this area.

04:38 Same sort of logic for the blue equation on the bottom.

04:42 That's the lower bound...

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