Maybe it's a crude approximation, but it makes for an easy calculation of area. Area via a left Riemann sum. As you move the pink points, a rectangle is highlighted, and the calculation of its area is shown in the upper right corner.
The true area through the right of the highlighted rectangle is calculated along with the error between the true area and the corresponding area calculated with the Riemann sum. More information about applet. This estimate should agree with what you calculate with the above applet for that function and four subintervals.
OK, it's a sum over an infinite number of terms, whatever that means, but let's not get hung up on that. We call this Riemann sum a left Riemann sum.
How does the right Riemann sum compare to the left Riemann sum? The below applet will let you experiment. Does this number seem to be the same as with the left Riemann sum? Area via a right Riemann sum. This method for computing area should seem familiar.
And this was actually the first example that we looked at where each of the rectangles had an equal width. So we equally partitioned the interval between our two boundaries between a and b. And the height of the rectangle was the function evaluated at the left endpoint of each rectangle.
And we wanted to generalize it and write it in sigma notation. It looked something like this. And this was one case. Later on, we looked at a situation where you define the height by the function value at the right endpoint or at the midpoint. And then we even constructed trapezoids. And these are all particular instances of Riemann sums. So this right over here is a Riemann sum. And when people talk about Riemann sums, they're talking about the more general notion.
You don't have to just do it this way. You could use trapezoids. You don't even have to have equally-spaced partitions. I used equally-spaced partitions because it made things a little bit conceptually simpler. And this right here is a picture of the person that Riemann sums was named after. This is Bernhard Riemann. And he made many contributions to mathematics. But what he is most known for, at least if you're taking a first-year calculus course, is the Riemann sum.
And how this is used to define the Riemann integral. Both Newton and Leibniz had come up with the idea of the integral when they had formulated calculus, but the Riemann integral is kind of the most mainstream formal, or I would say rigorous, definition of what an integral is.
So as you could imagine, this is one instance of a Riemann sum. We have n right over here. The larger n is, the better an approximation it's going to be. How can we refine our approximation to make it better? The key to this section is this answer: use more rectangles. The Left Hand Rule says to evaluate the function at the left-hand endpoint of the subinterval and make the rectangle that height.
In Figure 1. The Right Hand Rule says the opposite: on each subinterval, evaluate the function at the right endpoint and make the rectangle that height. The Midpoint Rule says that on each subinterval, evaluate the function at the midpoint and make the rectangle that height.
These are the three most common rules for determining the heights of approximating rectangles, but we are not forced to use one of these three methods. Interactive Demonstration.
The areas of the rectangles are given in each figure. Figure 1. In this figure, these rectangles seem to be the mirror image of those found in Figure 1. This is because of the symmetry of our shaded region. Our approximation gives the same answer as before, though calculated a different way:.
The notation can become unwieldy, though, as we add up longer and longer lists of numbers. We introduce summation notation also called sigma notation to solve this problem. Do not mix the index up with the end-value of the index that must be written above the summation symbol. The index can start at any integer, but often we write the sum so that the index starts at 0 or 1. The output is the positive odd integers. Evaluate the following summations:.
The following theorems give some properties and formulas of summations that allow us to work with them without writing individual terms. Examples will follow.
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