Riemann Sums, Trapezoidal Rule, and Simpson’s Rule Haley Scruggs 1 st Period 3/7/11

A Brief Overview of what Riemann Sums, Trapezoidal Rule and Simpson’s Rule do… Each of these method estimate the area of a curve using rectangles. As your number of rectangles increase so does the accuracy of the area.

A Refresher On Left Riemann Sums The (b-a) is the interval in which the area estimation is being calculated. N is the number of rectangles B is the higher number in the interval and A is the lower.

Left Riemann Sum Cont. To get your different x’s, you must first add to the A until you get B. Note that the number of rectangles you are given is the number of times you must add. After getting your different x’s you then plug them back into the original equation and then use the Riemann Sum Equation to find your estimated area.

Lets See an Example * Note that in Left Riemann Sum you use the first number in the interval but not the last when finding the area. This is because you are coming from the left side of the curve not the right*

Example Cont. * Plug these numbers back into the original equation*

Next Up on the Refresh Train is Right Riemann Sums The only real difference in the equation for Right Riemann Sum is the fact that you are going to use the last number in the interval but not the first. This is because you are coming from the right side of the curve instead of the left.

Lets see that Example *Remember you are not going to use the first number in the interval but you will use the second number in the interval when calculating your estimated area. *

Example Cont.

Finally the last Riemann Sum is Midpoint There is a slight difference between Midpoint Riemann Sum and the other two Riemann Sum techniques. The first is that you use both of the numbers in the interval to find your final answer. The other difference is that you divide the x’s by two and then plug that into the f(x) equation to find your area.

Now for the Example *After you add your ¼ to all the numbers, go back and add the number you added the ¼ to and the new number together and divide by two then plug that number into the equation.* These are the numbers you plug into the equation.

Example Cont.

Alright next up is Trapezoidal Some slight differences in the Trapezoidal Rule equation to notice. First when finding the area you now have a 2n instead of the original n. The other thing to notice is that you have to multiply the f(x) by 2, except for the first and last f(x).

Trapezoidal Cont Note that even though the multiplier out front is you still will use to find what your x’s. The number of rectangles is still used as a check to make sure your math is correct when finding your x’s, too. You also use all of the x’s.

The Trapezoidal Ex There is more than one way to write the equation. The A is the lower number on the integration sign and the B is the higher number on the integration sign. You do not multiply the N by 2 to find your additive.

Example Cont. *Note the new multiplier on the outside.* *You have to multiply f(x) by 2 now except for on the first and last term.*

Finally on the Review list is Simpson’s Rule There are three differences between Simpson’s Rule and the Trapezoidal Rule.

Simpson’s Cont. 1. There is now a 3n on the bottom instead of a 2n. 2. You have to alternate between 2 times and 4 times the f(x) instead of the constant 2 times f(x). 3. Simpson’s Rule can only be used for an even number of rectangles.

Simpson’s Example You do not multiply the N by 3 to find your additive. As can see the problem is started off the same way as all the others you will see the differences in later steps

Example Cont.

Now for the easier way to do these problems… The Calculator Method! There are two calculator methods. The first will give you Trapezoidal and all the Riemann Sums. The second will give you Simpson’s Rule.

Now for the Steps on how to do it Turn the calculator on and go to Programs Scroll down until you see Riemann and hit enter then hit enter again. Once there enter your function and press enter, enter your interval and hit enter, and finally enter your number of rectangles and hit enter. Once it is done graphing your function, hit enter and choose which method you would like it to use and hit enter. It should pop out an answer. Turn on the calculator and go to Y= and put in your equation. Then go to Programs. Scroll down till you see SIMP then hit enter. Put in the number of rectangles and hit enter. Put in what A = and hit enter Put in what B= and hit enter It should spit out an answer based on the equation you put in your y=.

Calculator Try Me Left Riemann Right Riemann Midpoint Riemann Simpson’s Rule Trapezoidal Rule [0,2] n=4 Answers: Left Riemann = Right Riemann =.2601 Midpoint =1.375 Simpson’s = Trapezoidal=

Now for the Try Me Problems Left Riemann

Try Me: Left Riemann Answer

Try Me: Right Riemann

Right Riemann Answer

Try Me: Midpoint

Midpoint Answer

Trapezoidal Try Me

Trapezoidal Answer

Simpson Try Me

Simpson Answer

One Last Thing… An FRQ 2004 (Form B) AB3 T(min) V(t) (mpm) A. Use a midpoint Riemann sum with four subintervals of equal length and values From the table to approximate Show the computations that lead to your answer. =229 © Haley Scruggs 2011