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Sketch the region enclosed by the given curves an…

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Problem 16 Easy Difficulty

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

$ y = \cos x $ , $ y = 2 - \cos x $ , $ 0 \le x \le 2\pi $

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Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 6

Applications of Integration

Section 1

Areas Between Curves

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Video Transcript

Okay, this question: we need to sketch the region closed by those 22 curves and to find its area. So here is our x y plane and that, as we see, the domain is from 0 to 2 pi to a better focus on the right side to 2. Pi the first side, cotingas at looks like this. We know x, equals to pi over 2 cos x, equals to 0 and x, equals to 0 cos x equals to 1. So here is 1, so it goes like this here, it's 91 this is x, equals. Then it goes to y again. This will be all the cosine x and 2 minus cosine x, 2 minus cosine x. I i the x, equals to 0 point. This equals to 1. Okay have x, goes to pi this actually equal to 3. So this is 3. So it's like the flip, this cosine x y, this y equals 1 and then we'll get our 2 minus cosine x. This function looks like this: it reach to 3 that x equals to y. Then it goes down to okay. Then we need to find area between. Okay, so we see all the things represented by x. We determine in to use x as our integration factor, so we take the integral with respect to x and a young to x goes from 0 to 2. Pi here is just a thing inside the integral is upper curve, is minus the lower curve, so is 2 minus cos x, minus cosine x, so it will be 2 minus 2 cosine x and the anti derivative of this will be 2 x. Minus 2 sine x, evaluate x goes to 2 pi e goes to 0 point over x, equals 2 pi is equals to 4 pi sine 2 pi 0. So this term cal, so we got 4 pi and minus x equals to 0 point. We know everything here is 0, so our answer will before pi.

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Calculus: Early Transcendentals

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