Find the exact area under the cosine curve y=cosx from x=0 to x=b, where 0⩽b ⩽π/ 2 . (Use a comp

Find the exact area under the cosine curve y=cosx from x=0...

Key Concepts

Definite Integral

A definite integral calculates the signed area under a curve over a specific interval. It is a core concept in integral calculus that involves computing the accumulation of quantities, where the integral from a to b gives the net area between the function and the horizontal axis from point a to point b.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, showing that integration can be reversed by differentiation. It states that if a function is continuous over an interval, then its definite integral can be computed using any of its antiderivatives; this is key to evaluating the exact area under curves.

Riemann Sums

Riemann sums are a method of approximating the area under a curve by summing the areas of a sequence of rectangles. As the number of rectangles increases (and their width decreases), the Riemann sum converges to the exact value of the definite integral, illustrating the limit process that underpins integral calculus.

Integration of Trigonometric Functions

Integration of trigonometric functions, such as the cosine function, involves applying antiderivative rules specific to trigonometric identities. By knowing the standard antiderivatives of sine and cosine, one can exactly evaluate the area under these curves, making it a fundamental skill in both theoretical and applied calculus.

Transcript

00:01 All right, so today we're going to be finding the exact area under a given function from one start point to an endpoint.

00:07 So our given function, f of x, our given function is cosine of x.

00:15 And we know that we're going to be starting, so our start point, which x equals a.

00:19 Our start point is actually equal to zero.

00:21 We're going from our start point to our endpoint.

00:24 Our endpoint is x equals b.

00:26 So we're actually going to be solving with the endpoint x equals b.

00:29 But i'm also going to do an example at the end using so the example at the end we're going to be using b equal to pi over 2 now the last piece of information that's given in the in our question is that b is inside the continuous range so it could be these values of pi over 2 and 0 so it's somewhere in between that and it can be equal to 0 or pi over 2 so graphically if we look at this the question is asking is given the function cosine of x which is that blue graph right there it wants us to find the exact value from here zero all the way to a value of b there's b and we know that b could actually go from any way for in zero to pi over two so b could go from pi over two all the way to zero and we just want to find the exact area so normally the way you do this is you use your rectangles and everything and you calculate them but we want the exact value so we can't use a remand sum like that for an estimation so the way we use the exact value is we're going to take that exact area we're actually going to find the sum of all those little rectangles right all the little rectangles we're going to find the infinite sum of them all so imagine if you had infinity rectangles and you were adding all that area up and the way you do this is with this given formula right here so you take your function your f of x value right take your function and depending on the height and the number the rectangle number right so each rectangle in their height that will be this value then you've got to multiply by the thickness of those rectangles then you're adding all of them up so all your rectangles infinite of them so now if we relate this general equation right this general equation this one right here if we relate this general equation general equation relate that general equation to the given question we have still that infinite sum of all the rectangles.

02:41 Now we know our function.

02:42 We know that our function is cosine of xi.

02:50 So let's break this general equation up until our different components.

02:54 Start with the thickness.

02:56 So this is actually given by the equation of b a over n.

03:00 That's how you find the thickness of all your rectangles.

03:03 So let's look at it in this case.

03:05 We know that b, it's still b.

03:07 A is now zero.

03:09 And it's still n.

03:10 So delta x.

03:12 It's actually equal to b.

03:13 Over n so we can substitute that into our equation now let's look at the value of x i star so x i star can normally be found for any sample point right any sample point x i is equal to zero plus i times the thickness of your rectangle so if we plug in our numbers we can find this is actually equal to plus which we just found right be over n we just found out all we're doing is plugging it in there b over n so x i is equal b i over n now if we take both of these values and plug it into our general formula one more time we get that the exact area is equal to the limit is n approaches infinity of the summation starting at number one going all the way till infinity right that's what we have in infinity of the cosine of b i over n times the quantity of b over n...

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