Sketch the region enclosed by the given curves and find its area. y=cosx, y=1-cosx, 0 ⩽x ⩽π

Name: Problem 24, Chapter 5: Calculus
Uploaded: 2017-11-14T06:35:40-08:00
Duration: 18 min 46 s
Channel: Apoorva Shah
Description: Sketch the region enclosed by the given curves and find its area. y=cosx, y=1-cosx, 0 ⩽x ⩽π

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

Key Concepts

Area Between Curves

The area enclosed by curves is calculated using the integral of the difference between the upper and lower function over the interval defined by their intersections. This concept is central to problems in calculus where determining the exact area between non-linear functions is required.

Trigonometric Integration

Computing the integral of trigonometric functions precisely requires familiarity with integration techniques specific to these functions. This includes knowing antiderivatives of sine and cosine functions and applying appropriate integration limits, which are crucial when evaluating the area bounded by trigonometric curves.

Graphing Trigonometric Functions

Understanding the general shape and transformations of trigonometric functions, such as the cosine function and its vertical translations, is fundamental. This includes knowing the period, amplitude, and key features like intercepts, maxima, and minima, which helps in accurately sketching the curves and visualizing the region they enclose.

Intersection of Curves

Finding the points where two curves intersect involves solving the equation that equates the functions. This process determines the limits of integration and is essential for accurately partitioning the region of interest when the top and bottom functions switch roles.

Transcript

00:04 Okay, so the first part of this problem asks us to sketch the region and close by these given curves.

00:12 So we'll start with that.

00:17 So here's what our axis will look like.

00:22 And we'll start with cosine of x.

00:25 And this, hopefully you're familiar with.

00:28 So we'll just mark down some key points.

00:32 We'll start with pi over here, and then pi over two.

00:37 So then you know some values of cosine, for example, cosine of 0, 0, or that equals 1, cosine should note these are.

01:28 Okay, cosine of pi over 2, 2 equals 0.

01:51 And then finally, we have cosine of pi, and that equals negative 1.

02:15 So we're going to use these values to also help us draw our graph for the other function, as you'll see.

02:22 So first we have these values, so we can plot the graph.

02:28 We have this, this, and we know the general shape of the cosine graph.

02:39 So it'll look something like that.

02:44 Okay.

02:47 Now we want to draw our graph of 1 minus cosine of x.

02:51 And since we've already figured out these values, we can already fill in most of it.

02:57 So for 1 minus cosine of x, we have at 0, at pi over 2, that that would be 1 minus pi over 2.

03:13 1 minus 1, cosine of x is 1.

03:16 So that is this.

03:19 And finally, at pi, at pi we have that it's 1 minus negative 1.

03:35 So that is 2.

03:46 And i'll specify that this is negative, or this is 2.

03:57 Okay, so now let's fill in this graph.

04:00 And again, think about what this graph should look like.

04:03 Well, it's the cosine graph, but it's flipped, and then it's shifted up once.

04:08 So this general shape will still be that of a cosine graph, but it's flipped.

04:13 So it'll look something like...

04:19 I just want to emphasize the curves a little bit more, so i will redraw this.

04:28 It's a little bit more like that.

04:31 Okay, so now that we have the graphs sketched, we want to find the area of this region over here, and this region.

04:59 Okay, so this means we have to first figure out what this point right here is, because we have to integrate from here to here, and then we need to switch our graphs and integrate from here to here...

Please give Ace some feedback