(a) Use six rectangles to find estimates of each type for the area under the given graph of f fr

(a) Use six rectangles to find estimates of each type for...

Key Concepts

Left Endpoint Approximation

The left endpoint approximation is a type of Riemann sum where the height of each rectangle is determined by the value of the function at the left endpoint of the subinterval. This method is straightforward to compute but can lead to systematic overestimation or underestimation of the actual area, depending on whether the function is increasing or decreasing on the interval.

Riemann Sums

Riemann sums are a fundamental concept in integral calculus, used to approximate the area under a curve. They work by partitioning the interval of interest into smaller subintervals, then summing up areas of simple shapes (usually rectangles) whose heights are determined by the function’s value at specific chosen points within each subinterval. This approach provides an estimation of the definite integral, and as the number of subintervals increases, the approximation generally improves.

Right Endpoint Approximation

The right endpoint approximation uses the value of the function at the right endpoint of each subinterval to determine the height of the corresponding rectangle. Like the left endpoint method, this approximation can also cause consistent overestimation or underestimation based on the function’s behavior, which directly impacts how close the approximation is to the true area under the curve.

Midpoint Approximation

The midpoint approximation improves upon the left and right endpoint methods by using the value of the function at the midpoint of each subinterval as the height of the rectangle. This method often provides a more balanced estimate because it tends to average out the errors that occur from sampling only one endpoint in methods where the function is either increasing or decreasing.

Overestimate and Underestimate in Numerical Integration

In numerical integration, whether an approximation yields an overestimate or an underestimate depends on the behavior of the function over the interval. For example, if a function is increasing, the left endpoint approximation will tend to underestimate the area while the right endpoint will overestimate it. Understanding these tendencies is crucial for analyzing the error in an approximation and for choosing a method that best suits the characteristics of the function.

Transcript

00:01 All right, hello, so today we are working with riemann sons, and we are trying to find the area under the given graph of f, which is shown here.

00:09 So let's get started.

00:12 So to start, we are working with the left end points.

00:16 And so first, we have to find the width of each of our rectangles.

00:22 So we know that we are making six rectangles.

00:26 And here we go.

00:28 So the equation that we want to use is delta x, which to know.

00:31 How wide each of the rectangles is going to be is equal to b minus a all over n and so b here is going to be 12 because this is the last x point that we are required to find a is going to be zero because that's where we are starting and n is the number of rectangles that we are using to find the area under the graph so once you plug all of these values in you have 12 minus zero which is equal to 12, divide that by 6, and you get 2.

01:06 So each of our rectangles is going to be two units wide.

01:10 And so i'm going to mark where each of our rectangles is going to end, so how wide each rectangle will be.

01:19 And let's start drawing out our rectangles.

01:22 So first we are going to start on the left end point, which is going to be the leftmost part of this two unit wide rectangle.

01:33 And we're going to start by making sure that you are on the graph and you're going to follow it out.

01:41 You're going to go straight over from where you first marked your pencil over two units and then you're going to follow it all the way down.

01:48 So i'm going to switch colors just so you guys can have a better visual and a clearer understanding of where the rectangle is starting to stop.

01:57 So first we're starting from the graph.

01:59 We're going out.

02:00 Don't follow the graph.

02:01 Just go straight out and then go all the way down.

02:06 So next we are starting there, going out two units, and going all the way down.

02:14 And you're going to do this for all six rectangles.

02:16 Going out, going all the way down.

02:21 Going out, going all the way down.

02:26 And here's our last rectangle, going out.

02:30 And remember that you're starting from the graph and going all the way down.

02:32 So here we have our six beautiful rectangles.

02:35 So now we are going to go to this formula, which is going to help us find the area under the graph.

02:44 So here we have l6, which is the left endpoints and that there are six rectangles is equal to the summation from i is equal to 1 all the way to 6, because that's how many rectangles we have.

02:59 The summation of f of x sub i minus 1 times delta x.

03:05 So we've already calculated here, delta x being equal to 2.

03:11 So we can just plug the 2 in and let's get started.

03:17 So once we start working with our formula, uh -oh, let's shift that over.

03:31 Okay, there we go.

03:32 Sorry, that's still a little off, but you get the general idea.

03:36 So once we start working with our formula, so we are going to put delta x in the beginning because we're multiplying every, all of this by delta x.

03:46 So the summation of this multiplied by delta x.

03:50 So we get two times f of x sub 0 plus f of x sub 1 plus f of x sub 2.

04:05 And we're going to go all the way to f of x sub 5.

04:10 All right.

04:11 And so this is then equal to two times f of zero plus f of two plus f of two plus f of four plus f of six.

04:28 Plus f of 8 plus f of 10.

04:35 And we got these numbers from each endpoint.

04:41 So this is 0.

04:46 This is 2.

04:47 So the endpoint of this second rectangle is at x is equal to 2.

04:52 The endpoint of the third rectangle is at x is equal to 4 and so on.

04:56 So that's where we get these numbers.

04:58 All right.

04:59 Now we just plug in the numbers at each point.

05:03 So this is, so an approximation because these are all estimates, because as you can see that these don't exactly follow the graph, which is why we can't say that it is the for sure area under the graph.

05:19 But now let's plug these numbers in.

05:22 So we know that f of zero is equal to 9, f of 2 is equal to 8 .8, f of 4 is equal to 8 .2.

05:31 F of 6 is equal to 7 .3, f of 8 is equal to 5 .9, oops, and f of 10 is equal to 4.

05:45 And so if we just plug these all into our calculator and solve it, we get 2 times 43 .3.

05:51 And we have estimated that the area under the curve using the left endpoints is 86 .6.

05:58 Six.

06:00 All right.

06:01 So i am going to use this same formula for the next problem.

06:10 So let's keep shugging along.

06:12 So next we are to use the right end points.

06:20 So let's see if this will work.

06:24 Oopsie.