What is the difference between right and left endpoints

Let's calculate the Left Riemann Sum for the same function. The left endpoints of the intervals are 0,1, and 2. So we evaluate f there:. So the heights of our rectangles are 1,2, and 5. The widths of the rectangles are still 1 unit each because we didn't change our intervals. Then the area is. Finally, the Right Riemann Sum uses the right-endpoints of the mini-intervals we construct and evaluates the function at THOSE points to determine the heights of our rectangles.

For our particular example, the right endpoints are 1,2, and 3. Then the total area is. You've now computed some simple Riemann Sums, of each of the three main types we want to talk about here. But this leaves a few questions unanswered. That's not going to be fun. And I doubt it's possible to compute exactly. Oftentimes in industry an exact value isn't necessary, and an approximation is enough, so one can do a calculation like the ones we just showed you. It's also worth noting that especially if you have big numbers or a lot of rectangles, these calculations are usually done on a computer.

Here's another question you might think about: how many rectangles do we need? We can use as many as we want, but as we add more, the widths of them will be smaller, and we will have to sum more areas.

If we get to using so many, we will get very close to the exact value of the integral. This is exactly why we care about limits, and it so turns out that the integral will equal the exact same thing as any of the Riemann Sums if we approach infinitely many rectangles and of course, the individual area of each rectangle will go to zero, but we gain more and more rectangles, so these effects essentially cancel each other out.

If you're looking for a big fancy formula, you won't have to use one in practice for this, but it's helpful for understanding what's going on.

Here's a formula for using Left Riemann Sums to find the value of the integral for a function f:. The exact locations are determined as a weighted average between the two endpoints of the big interval, a and b.

The last thing I want to talk about is how good our approximations are. That is, how close can we get to the actual value of an integral using a Riemann Sum? There are so many functions that it ultimately depends. I'm going to explore this from two different dimensions. But the expressions inside the limits are different, so what gives? How close? The endpoints of each subinterval are,. These points will define the height of the rectangle in each subinterval. Note as well that these points do not have to occur at the same point in each subinterval.

However, they are usually the left end point of the interval, right end point of the interval or the midpoint of the interval. We will use summation notation or sigma notation at this point to simplify up our notation a little. If you need a refresher on summation notation check out the section devoted to this in the Extras chapter. In other words,. Many functions are not positive however. Our answer is negative as we might have expected given that all the function evaluations are negative.

Now, what about a function that is both positive and negative in the interval? In fact that is correct. Here the area estimation for this case. Notes Quick Nav Download. We could evaluate the function at any point c i in the subinterval and use as the height of our rectangle.

This gives us an estimate for the area of the form. A sum of this form is called a Riemann sum , named for the 19th-century mathematician Bernhard Riemann, who developed the idea.

Recall that with the left- and right-endpoint approximations, the estimates seem to get better and better as get larger and larger. The same thing happens with Riemann sums. Riemann sums give better approximations for larger values of. We are now ready to define the area under a curve in terms of Riemann sums. Let be a continuous, nonnegative function on an interval and let be a Riemann sum for Then, the area under the curve on is given by. See a graphical demonstration of the construction of a Riemann sum.

Some subtleties here are worth discussing. First, note that taking the limit of a sum is a little different from taking the limit of a function as goes to infinity. Limits of sums are discussed in detail in the chapter on Sequences and Series in the second volume of this text; however, for now we can assume that the computational techniques we used to compute limits of functions can also be used to calculate limits of sums. Second, we must consider what to do if the expression converges to different limits for different choices of Fortunately, this does not happen.

Although the proof is beyond the scope of this text, it can be shown that if is continuous on the closed interval then exists and is unique in other words, it does not depend on the choice of. We look at some examples shortly. If it is important to know whether our estimate is high or low, we can select our value for to guarantee one result or the other. If we want an overestimate, for example, we can choose such that for for all In other words, we choose so that for is the maximum function value on the interval If we select in this way, then the Riemann sum is called an upper sum.

Similarly, if we want an underestimate, we can choose so that for is the minimum function value on the interval In this case, the associated Riemann sum is called a lower sum.

Note that if is either increasing or decreasing throughout the interval then the maximum and minimum values of the function occur at the endpoints of the subintervals, so the upper and lower sums are just the same as the left- and right-endpoint approximations. Find a lower sum for on let subintervals. With over the interval We can list the intervals as Because the function is decreasing over the interval Figure shows that a lower sum is obtained by using the right endpoints.

Find a lower sum for over the interval let. The intervals are and Note that is increasing on the interval so a left-endpoint approximation gives us the lower sum. A left-endpoint approximation is the Riemann sum We have. Using the function over the interval find an upper sum; let. Follow the steps from Figure. State whether the given sums are equal or unequal.

They are equal; both represent the sum of the first 10 whole numbers. They are equal by substituting d. They are equal; the first sum factors the terms of the second.

In the following exercises, use the rules for sums of powers of integers to compute the sums. Suppose that and In the following exercises, compute the sums. In the following exercises, use summation properties and formulas to rewrite and evaluate the sums. Let denote the left-endpoint sum using subintervals and let denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval.

L 4 for on. R 4 for on. L 6 for on. R 6 for on. L 8 for on. Compute the left and right Riemann sums— L 4 and R 4 , respectively—for on Compute their average value and compare it with the area under the graph of.

Compute the left and right Riemann sums— L 6 and R 6 , respectively—for on Compute their average value and compare it with the area under the graph of.

The graph of is a triangle with area 9. Compute the left and right Riemann sums— L 4 and R 4 , respectively—for on and compare their values. Compute the left and right Riemann sums— L 6 and R 6 , respectively—for on and compare their values. They are equal. Express the following endpoint sums in sigma notation but do not evaluate them.

L 30 for on. L 10 for on. R 20 for on. R for on. In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. Is the area under the curve between the left and right endpoint sums? The plot shows that the left Riemann sum is an underestimate because the function is increasing.

Similarly, the right Riemann sum is an overestimate. The area lies between the left and right Riemann sums. Ten rectangles are shown for visual clarity. This behavior persists for more rectangles. The left endpoint sum is an underestimate because the function is increasing.

Similarly, a right endpoint approximation is an overestimate. The area lies between the left and right endpoint estimates. Let t j denote the time that it took Tejay van Garteren to ride the th stage of the Tour de France in If there were a total of 21 stages, interpret.

Let denote the total rainfall in Portland on the th day of the year in The approach is a geometric one. By dividing a region into many small shapes that have known area formulas, we can sum these areas and obtain a reasonable estimate of the true area.

We can use this regular partition as the basis of a method for estimating the area under the curve. We next examine two methods: the left-endpoint approximation and the right-endpoint approximation. The second method for approximating area under a curve is the right-endpoint approximation. It is almost the same as the left-endpoint approximation, but now the heights of the rectangles are determined by the function values at the right of each subinterval.

We find the area of each rectangle by multiplying the height by the width. When the left endpoints are used to calculate height, we have a left-endpoint approximation. This is the width of each rectangle. We have. Approximate the area using both methods. See the below Media. We will have more rectangles, but each rectangle will be thinner, so we will be able to fit the rectangles to the curve more precisely.

We can demonstrate the improved approximation obtained through smaller intervals with an example. The area of the rectangles is.