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

Find the exact area under the cosine curve $y=\cos x$ from $x=0$ to $x=b$, where $0 \leqslant b \leqslant \pi / 2$. (Use a computer algebra system both to evaluate the sum and compute the limit.) In particular, what is the area if $b=\pi / 2$ ?

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

00:04 So problem 5 .1, number 31, so they want us to find the exact area under the cosine curve.

00:11 So y is equal to cosine x.

00:13 We're going to go from x equals 0 to k, and we're guaranteed that k is between 0 and pi over 2.

00:21 So in this case, we're going to use rectangles.

00:24 And so the width of a rectangle is going to be k minus 0 over n, because i have an rectangle.

00:34 So k over n is going to be the width of every one of these rectangles.

00:39 Now the other thing i know, i know that this value of k is between zero and pi over two.

00:45 So i know that y equal cosine x, that is greater than equal to zero on the interval, zero to pi over two.

00:56 So that means that when i graph this guy, and so you should know what the graph of this looks like from zero to pi over two, this guy starts out at one ends at zero so you've got something going on like this we are going to estimate with n rectangles okay in order to and then set the limit to get the exact value so i see that every rectangle is above the x -axis so i'm going to start using right in points so each in each width of each rectangle is k over n and then you're going to have the value of the cosine at that point so the first one is going to be cosine of pi over n, and then cosine, excuse me, pi over n.

01:45 And then from there, you'll have cosine of, and sorry about that with the k.

01:49 So it's k.

01:53 Oh, let me back up.

01:54 So sorry about that.

01:57 So when you do one, so it's going to be the cosine of k over n.

02:03 So cosine of k over n, the next one would be the cosine of k over n.

02:08 Twice that 2k over n, 3k over n until finally you get to get to k.

02:17 Okay, so in this case, so let's take a look at it.

02:21 What would be our sum? so the area is going to be the limit as the number of rectangles approaches infinity of the sum of all the areas of each rectangle...

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