Special types of Riemann sums

Transcription

1 Roberto s Notes on Subject Chapter 4: Deinite integrals and the FTC Section 3 Special types o Riemann sums What you need to know already: What a Riemann sum is. What you can learn here: The key types o Riemann sum that are preerentially used. In the construction o a Riemann sum, we have to choose which value o x to pick to compute the height o each approximating rectangle. It turns out that in the main application that will develop, such a choice does not matter. However, in other applications it may be an issue and all we need to do in this section is to assign names to certain special choices that are oten used and have particular eatures. We start rom two related choices that are made based on the position o the value in each interval. Knot on your inger: One-sided Riemann sums To estimate the area under a continuous unction y x ab, on the interval, we can use: the let Riemann sum by picking the let end value o x or each slice the right Riemann sum by picking the right end value o x or each slice. Example: I we want to construct a Riemann sum over the interval slices, we are dividing the interval into 6 parts deined by the smaller intervals 1,,, 3, 3, 4, 4, 5, 5, 6, 6, 7. Thereore: 1, 7 based on 6 or the let Riemann sum we pick all the let end values, thus getting: Integral Calculus Chapter 4: Deinite integrals and the FTC Section 3: Special types o Riemann sums Page 1

2 or the right Riemann sum we pick all the right end values, thus getting: Example: With the same setting as beore, the midpoint Riemann sum is given by: Both o these choices seem one sided (they are!), so an obvious alternative is to stay in the middle. Knot on your inger Midpoint Riemann sum To estimate the area under a continuous unction y x ab, on the interval, we can use the midpoint Riemann sum by picking the middle value o x or each slice The last two popular choices are directed by the unction itsel. Knot on your inger Extreme Riemann sums To estimate the area under a continuous unction y x ab, on the interval, we can use : the upper Riemann sum by picking the largest value o x or each slice the lower Riemann sum by picking the smallest value o x or each slice Integral Calculus Chapter 4: Deinite integrals and the FTC Section 3: Special types o Riemann sums Page

3 Example: With the same setting as beore, here is what the upper (on the let) and lower (on the right) Riemann sums would provide: All o these choices can be useul at some points, but it is important to keep the ollowing properties in mind. Quick portrait o Special Riemann sums The let Riemann sum: is easy to set up by hand, since the values o x are easy to identiy is an overestimate i is an underestimate i x is decreasing x is increasing The right Riemann sum: is easy to set up by hand, since the values o x are easy to identiy is an overestimate i is an underestimate i The midpoint Riemann sum: x is increasing x is decreasing is easy to set up by hand, since the values o x are easy to identiy is decent estimate in most cases o interest. The upper and lower Riemann sum: are not easy to set up by hand, since the maximum or minimum must be computed separately or each slice are poor approximations, since the upper is always an overestimate and the lower is always an underestimate, but... provide bounds or the possible values o the correct area. Can you give me a numerical example? But o course Integral Calculus Chapter 4: Deinite integrals and the FTC Section 3: Special types o Riemann sums Page 3

4 Example: This is the region bounded by the x unction y e e in the irst quadrant. To estimate its area by using 4 rectangles, we can use the let sum, which is also the upper sum, since the unction is decreasing. In that case we get: x 0, 0.5,1,1.5 i 1 A e e e e e e e e We can, alternatively, use a right sum, which is also a lower sum: x 0.5,1,1.5, i 1 A e e e e e e e e So, we know that the area has to be between 6.61 and By using a midpoint sum we get: x 0.5, 0.75, 1.5, 1.75 i 1 A e e e e e e e e This value is probably airly close to the exact value. By using the methods that you will learn in the next sections, we can actually arrive at a much more accurate estimate o Not to contradict what you say, but I got slightly dierent numbers rom those calculations. Assuming that you have pushed all the right buttons, most likely you used more or ewer decimal digits, thus getting dierent values. That is ine, since we are just computing approximations. In the next section you will see how to get away rom this problem (to a large extent) and arrive at an exact computation o these areas, at least theoretically. Summary Some special types o Riemann sums are used more oten than others, as they use particularly convenient choices o representative values. Common errors to avoid Do not conuse let and right sums with upper and lower sums: they are occasionally the same, but not always. Integral Calculus Chapter 4: Deinite integrals and the FTC Section 3: Special types o Riemann sums Page 4

5 Learning questions or Section I 4-3 Review questions: 1. Explain why it is useul to consider special types o Riemann sums.. Describe each special type o Riemann sum, together with its basic properties. Memory questions: 1. Which values are chosen or a let Riemann sum?. Which values are chosen or a right Riemann sum? 4. Which values are chosen or an upper Riemann sum? 5. Which values are chosen or a lower Riemann sum? 3. Which values are chosen or a midpoint Riemann sum? Computation questions: For each o the unctions provided in questions 1-5, construct the let, right, midpoint, upper and lower Riemann sums to approximate the area bounded by the unction and the x- axis between the two given values and with the given number o intervals. 1. y 0, 4, n=4. x on 4. y 1,.5, n=6. cosh x on. y sin x 3. y ln x on on 0, n= 6. 1, 6, n= y e x on 0, 5, n=5. Integral Calculus Chapter 4: Deinite integrals and the FTC Section 3: Special types o Riemann sums Page 5

6 Application questions: 1. I the velocity o a moving object is given by the unction seconds by using 6 rectangles and the midpoint sum. v t t 1 36, where t is in seconds and v in m/sec, estimate the distance travelled by the object in the irst 6 Templated questions: 1. Choose a unction that is positive on an interval o your choice and construct the special Riemann sums that approximate the area o the region it bounds. What questions do you have or your instructor? Integral Calculus Chapter 4: Deinite integrals and the FTC Section 3: Special types o Riemann sums Page 6