Riemann sums

So now that we're comfortable working with summation notation, we can turn toward Riemann sums which are a tool that we use to estimate the area under a curve. So let's think about this area problem here. When we have some function f of x, and let's say it's represented here by this curve. The integral as we've said is calculating the exact area under this curve. So the area enclosed by this curve and the x axis, this shaded region here that we've labeled with a for area. And if we're just taking an indefinite integral, then we don't limit the area of this shaded region to this left edge here at x equals a and this right edge here at x equals b, just goes on forever infinitely to the left and right. And the integral, the indefinite integral calculates the area under the curve. When we work with a Riemann sum, what we're doing is we're approximating this shaded region. We're trying to find an estimate of the area. The idea of a Riemann sum is to slice this shaded region into many thin rectangles. So think about a vertical rectangle that gives the area of just this first little strip. We know easily how to calculate the area of a rectangle, it's just length times width. So we would take this tiny little width and the length which in this case is the height or the distance from the x axis to the value of the curve here and we calculate the area of that rectangle. And then we take another tiny little rectangle and we calculate its area. And we keep slicing this whole area into little rectangles, calculating the area of each rectangle and then adding up all those areas to get an estimation of total area under the curve. Now the more rectangles we use, the better our approximation is going to be. And with Riemann sums, and this is also the case with integrals, when the function f encloses area that is above axis, we treat that area as positive. If the function dips below the x axis and therefore encloses area between the curve and the x axis that's below it, treat that area as negative. So what that means is that if we get a positive result for area, it tells us there's more area enclosed above the x axis than below it. And if we get a negative result for area, that means there's more area enclosed below the x axis than there is above it. Now when we're using a Riemann sum, there are three different approximations that we can make for area. The first one is a left endpoint approximation. So we've zoomed in here to the curve and the area under the curve. And what we can see is that we've used all these smaller rectangles to kind of approximate area. And the height of each rectangle is found at the left edge of the rectangle. So look at this biggest rectangle here on the right. The dot, the point where it intersects the curve, is on the left side of the rectangle. So this is a left endpoint approximation of area under the curve. When we're using a left Riemann sum or a left endpoint approximation, we use this formula here to estimate area. When we use a right endpoint approximation or a right Riemann sum, we use this formula here and when we use a midpoint approximation or a midpoint Riemann sum, then this is our area formula. Now we're going to go through some problems step by step here, but in general, this is what the Riemann sum is doing. To estimate area, we're adding up the area of a bunch of rectangles. Remember that we can find the area of each rectangle as width times height. Well this delta x right here is the width of each rectangle and this f of x sub I is the height of each rectangle. So this sum really is saying take the height of each rectangle, multiply it by the width of each rectangle, that's going to give us area of one rectangle. Start with the I equals first rectangle and go all the way to the nth rectangle, the last rectangle. So add up all the areas of the individual rectangles, and that's going to give us the area approximation. And that sum expanded just looks like finding the height at each endpoint, adding up all the heights, and then multiplying by this consistent rectangular width. So looking at this diagram here of the left endpoint approximation, here's how we can see that visually. Using these left endpoints here, the height of each rectangle is given by this length here, then this length, then this length, then this length, all the way up to the last rectangle. So we have the left side of each rectangle. Those heights are here, here, and here all the way up to the last rectangle. So we take all those heights, we add them up, then we multiply by delta x which is that width And the width here, delta x, is just this same consistent width right here, the width of each rectangle. And regardless of whether we're doing a left, right, or midpoint approximation, the concept is always the same. Add up all the heights, multiply by the width, and we'll get that area estimation. So let's look at an example. We've been given this function here g of x and we're asked to approximate the area under the curve on the interval two to eight with three equal subintervals or another way of putting that is three rectangles that have equal width. And we want to estimate area using left endpoints, right endpoints, and midpoints. So that being said, here's the step by step process that we'll use every single time to calculate a Riemann sum. The first thing we need to do is find delta x, the width of the rectangles. And we do that with this formula here. We say delta x is going to be equal to, we take the width of the entire interval. So we're looking at the interval here two to eight. So to find its width, we just take eight minus two, that's the width of the entire interval. Then we're going to divide that by the number of rectangles we're using or the number of equally wide subintervals which in this case we've been told is n equals three. This formula here is just b minus a divided by n or the width of the interval divided by the number of rectangles. So in our case that's six divided by three which is two. So the width of each rectangle is gonna be two units in this case. Then we wanna divide our sub interval into rectangles each with width delta x. So we're looking at this interval here two to eight. So we'll say two all the way up to eight. Each rectangle is gonna have width two, which means the first rectangle is gonna span from two to four. And then the second rectangle is gonna span from four to six. And then the last rectangle is gonna span from six to eight. And there are our three rectangles, n equals three, one, two, and three rectangles. Now we've been asked to use left endpoints, right endpoints, and midpoints. We're gonna do all three calculations in the same example. So if we picture these as our three rectangles, we can see that left endpoints of these rectangles are going to be two, four, and six leaving out this value eight. But if we use right endpoints, the right side of each of these three rectangles is four, six, and eight. And if we use midpoints, then we're looking at the value in the middle of each rectangle. So for this rectangle that spans two to four, the midpoint is three. The middle of the rectangle that goes from four to six is five, and the middle of the rectangle that goes from six to eight is seven. Now we've identified all of the points we'll use to plug into g of x to find the height of the curve at each of those points. Remember we already know width delta x equals two. Rectangles are always just height times width. So all we have left to do is calculate the heights at all these points here and then we'll have everything we need to get our area estimates. So our next step is to evaluate the function g at two, four, six, eight, three, five, and seven. Those are all the values we need. So let's just look at one example. So if we find g of two, we plug two into the right hand side here, get two cubed is eight, eight times a negative one half is negative four, two squared is four, four times five is twenty, so plus twenty, two times three is six so we get a minus six and then a minus eight and the result there is a positive two. So g of two is two. So the value of the function g at x equals two or the height of the rectangle at x equals two is two. So let's go ahead and put that here next to that value. G of four is twenty eight. If we plug four into g, we get twenty eight. If we plug in six, we get forty six. So the height there is forty six. This is twenty eight. This is forty six. And g of eight is thirty two. G of three is twenty nine halves. At x equals five we find seventy nine halves and at x equals seven we find eighty nine halves. So we calculate each of those heights. And now we can say that the left endpoint approximation, so we'll say here the left endpoint approximation of the Riemann sum with r equals three rectangles is going to be equal to the value of delta x, the width of the rectangles two multiplied by each of the heights that we found for these left endpoints here. So g of two, g of four, and g of six or two, twenty eight, and forty six. So two plus twenty eight plus forty six and the result there is a value of one hundred fifty two. So left endpoints with three rectangles estimate area under the curve to be one hundred and fifty two square units. If we look at right endpoints and midpoints, the Riemann sum approximation is going to be equal to for right endpoints, again, always have delta x equals two and then right endpoints here we have twenty eight plus forty six plus thirty two. So twenty eight plus forty six plus thirty two. And that's going give us an approximation of two twelve. And then midpoints with three rectangles is going to give us delta x, the width times the sum of all the heights. So we'll take twenty nine halves plus seventy nine halves plus eighty nine halves and the result there is one hundred ninety seven. Now if we actually integrate the function g, if we actually took the integral here of this function and we calculated area exactly, the value we would get there is one hundred and ninety two. And so if we plot these estimations and exact area on a number line, what we see is that we get a left endpoint approximation at one hundred fifty two. Let's maybe put that here at one hundred and fifty two. This is the left endpoint approximation. The midpoint approximation is one hundred ninety seven. We'll maybe put that here, one hundred ninety seven. That's midpoint. The right endpoint is two twelve. So that's up here, Two twelve, that's the right endpoint. And then exact area is at one hundred ninety two. So we'll put that here, one hundred ninety two. And that is what we'll call exact. What we can see here is that for this particular function g of x, left endpoints underestimate actual area, right endpoints overestimate actual area, midpoints still do overestimate actual area, but they get by far the closest to exact area and that will usually be the case. Midpoints are usually going to do a better job estimating area under the curve compared to left or right endpoints. Now the last thing we want to do here is say that we don't always have to have the function g in order to calculate a Riemann sum. We can also do this from a table. So let's say that instead of this function all we're given is a table of values for g at these values of x, x one to eleven and we're asked maybe to use a right Riemann sum with n equals five. So let's say that we're doing a right Riemann sum and we have n equals five and we want to estimate the area under g of x on the entire interval shown in the table from x equals one to x equals eleven. So as always with a Riemann sum we start by finding delta x. So delta x is always equal to b minus a divided by n or in this case eleven minus one, the width of the interval, divided by n, the number of rectangles and we can see that that's ten divided by five or two. So again our width is two. That means then that our first rectangle is going to span the interval one to three and therefore that the right endpoint of that rectangle is going to be at x equals three and therefore that the height at x equals three is going to be this value here. The height of the rectangle from one to three is five. The next rectangle is going to go from three to five. The right endpoint of that is going to be five. So this is the height there and if we keep going, these are all of our right endpoints twenty nine, fifty three, and eighty five. Those are the five heights for the n equals five rectangles. Which means then that the right Riemann sum with five rectangles is going to be equal to delta x two multiplied by g of three plus g of five plus g of seven plus g of nine plus g of eleven or simply these heights here, five plus thirteen plus twenty nine plus fifty three plus eighty five. And when we do that math, we find an area approximation of three seventy square units. And we can see how we can get that Riemann sum estimation from a table or directly from the function itself like we did in the previous example.