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Sketch the region enclosed by the given curves and find its area.
$ y = \cos \pi x $ , $ y = 4x^2 - 1 $
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Calculus 2 / BC
Applications of Integration
Areas Between Curves
Missouri State University
University of Michigan - Ann Arbor
University of Nottingham
Sketch the region enclosed…
Sketch the region bounded …
So you need to sketch the region enclosed by those votarine. Curves say this is our x y play and so the first function is cosine pi x, that means x equals to plus n pi equals to negative 1 point. So have letter x equals to 1 half cosine packs equal to 0 and the x equals to 0 co. Sine 0 is pi, because it's 0 is 1 point right. This function should look like and this is a even function. So its symmetric by y axis looks like this: is x equals 1. This is negative. 1. Excuse me, this is x, equals to negative 1 and y equals to negative 1. Second, and the second curve is a problem: not x, equals 0. This curve across y equals 1 axe 0, like both negative 1. Across this point, and when y equal to 0, you know x equals to positive and negative 1 half, so it cross those 2 points in it isn't native half right. So this problem. I look like this and the enclosed region is in between this shaded area. So, in order to find this area, we write on integral form the area of integral with respect to retuse x, because those functions can be represented by x explicitly so this is 8 was to integral, with respect with respect to x and x goes from negative 1 Half to positive 1 half- and here is the upper curve minus the lower this cosine x, minus 4 x square minus 1 point. So it will be minus 4 x, squared plus 1, all right next, when you to evaluate this definite integral this definite integral. So, let's find anti derivative, the first term there will be 1 over pi sine pi x, and the second term will be 4 over 3 x cubed. The third time, just where we value at our half negative or half the first term at x, equals to half know sine pi over 2 is 1 is 1 over pi minus 4 over 3 plus 1 poi minus plugging x, equal to negative, but half. So, as you can see, those are odd functions. Therefore, everything flap their sign, so when the here is the first term will be negative. 1. Half negative, sorry mistake: we plug in x equals 1 half here, so this is not 4 over 3 times 1. This 4 over 3 times 1 over 8 point- and here is just a plus 1 half right and the second term just flap the sign o the second. This is minus 1 over pie and plus 43 times tone over 8 minus or half right, and since we have this negative sign, if we breed here, everything we'll put are, therefore we will get 2 times the first first terms. We only need to calculate the first term and are multiplied by 2 right, so here will be over 6 plus 1 half that's or 3, so you got 2 over pi, plus 2 thirds.
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