Riemann Sum, Trapezoidal Rule, and Simpson's Rule

Riemann Sum, Trapezoidal Rule, and Simpson's Rule

RIEMANN SUM, TRAPEZOIDAL RULE, AND SIMPSONS RULE Cameron Clary Riemann Sums, the Trapezoidal Rule, and Simpsons Rule are used to find the area of a certain region between or under curves that usually can not be integrated by hand. RIEMANN SUMS Riemann Sums estimate the area under a curve by using the sum of areas of equal width rectangles placed under a curve. The more rectangles you have, the more accurate the estimated area.

Riemann Sums are placed on a closed integral with the formula: b a f ( x0 ) f ( x1 ) f ( x2 )... f ( xn ) n The interval is [a,b] and n is the number of rectangles used b a Is also called x and refers to the width of the rectangles x and refers to the width of the rectangles n f ( x0 ) f ( x1 ) f ( x2 )... f ( xn ) Represents the height of the rectangles There are three types of Riemann Sums: Left Riemann, Right Riemann, and Midpoint Riemann The left, right, and midpoint refer to the corners of the rectangles and how they are placed on the curve in order to estimate the area.

LEFT RIEMANN Left Riemann Sums place the left corner of the rectangles used to estimate the area on the curve. Left Riemann sums are an underestimation of the area under a curve due to the empty space between the rectangles and the curve. RIGHT RIEMANN Right Riemann sums place the Right corner of the rectangles on the curve. Right Riemann Sums are an overestimation of area because of all the extra space that is not under the curve that is still calculated in the area because it is inside the rectangles

MIDPOINT RIEMANN Midpoint Riemann Sums place the middle of the Rectangle on the curve Midpoint Riemann Sums are the most accurate because the area found in the part of the rectangle that is over the curve makes up for the area lost in the space between the curve and the rectangle LETS TRY AN EXAMPLE PROBLEM WITH LEFT, RIGHT, AND MIDPOINT RIEMANN SUM f ( x) x 2 On the interval [0,1] with n=4 First, find the width of the rectangles or x and refers to the width of the rectangles x

b a 1 0 1 n 4 4 Then, starting with a, the first number on the interval, plug the numbers into the f (x) x2 formula, adding x and refers to the width of the rectangles x each time. So x0 0 1 4 1 x2 2 3 x3 4 x4 1 x1

*You should always begin with a and end with b, if not, you plugged in the numbers wrong Once you have the numbers b a 1 n 4 and x0 0 You can plug them into the formula: b a 1 1 3 f (0 ) f ( ) f ( ) f ( ) n

4 2 4 When doing a Left Riemann, plug all numbers into the formula except for the last number. When doing a Right Riemann, plug in all numbers into the formula except for the first number. When doing a Midpoint Riemann, average the numbers and then plug in those values to the formula. 1 x1 4 1 x2 2 3 x3 4

x4 1 LEFT RIEMANN 1 1 1 3 f (0) f ( ) f ( ) f ( ) 4 4

2 4 Then, plug the 0, , , and into f ( x) x 2 so you get 1 1 1 9 7 2 0 units 4 16 4 16 32 RIGHT RIEMANN 1 1 1 3 f ( ) f ( )

f ( ) f (1) 4 4 2 4 Then, plug the , , , and 1 into f ( x) x 2 so you get 1 1 1 9 15 2 1

units 4 16 4 16 32 MIDPOINT RIEMANN 1 1 3 5 7 f ( ) f ( ) f ( ) f ( ) 4 8 8 8 8 Then, plug the , , , and 1 into f ( x) x 2 so you get 1 1 9 25 49 25

2 f ( ) f ( ) f ( ) f ( ) units 4 64 64 64 64 64 If the area can be found by hand, you can compare your answers from the Riemann Sums to the actual answer to see how accurate your estimation was In this problem, we can find the area by hand. The area for this problem is: 1 1 2 [ x ] dx units

3 0 2 Comparing the answers, the area found using the Left Riemann was under the amount of the actual area, the Right Riemann was over the amount of the actual area, and the Midpoint Riemann was the closest to the actual answer. None of the Riemann Sum types gave the exact answer, but that is because they are estimations. TRY THIS PROBLEM! Calculate the Left and Right Riemann Sum for on [0, ] using 4 rectangles.] using 4 rectangles. x0 0 b a n 4

x1 4 x2 2 3 x3 4 x4 f ( x) sin x ANSWERS 3 LR f (0) f ( ) f ( ) f ( ) 4 4 2 4

2 2 2 2 LR 0 1 1.8961 units 4 2 2 4 3 RR f ( ) f ( ) f ( ) f ( ) 4 4 2 4

2 2 2 RR 1 0 1.8961units 2 4 2 2 4 TRAPEZOIDAL RULE Trapezoidal Rule is very similar to the Riemann Sums, but instead of using rectangles to approximate area, it uses trapezoids. The trapezoidal rule is more accurate than the Riemann sums. TRAPEZOIDAL RULE When using the Trapezoidal Rule, use the formula: b a f ( x0 ) 2 f ( x1 ) 2 f ( x2 ) ... f ( xn )

2n The reason all but the first and last functions are multiplied by two is because their sides are shared by two trapezoids. b a n Is still added to each xnumber like in the Riemann Sums EXAMPLE PROBLEM Use the Trapezoidal Rule to Calculate f ( x) x 2 on the interval [0,1] when n=4 b a f ( x0 ) 2 f ( x1 ) 2 f ( x2 ) ... f ( xn ) 2n x0 0

x and refers to the width of the rectangles x= b a 1 n 4 1 x1 4 1 x2 2 3 x3 4 x4 1 b a 1 2n 8

Once you have all this information, all you have to do is plug the numbers into the formula f ( x) x 2 [0,1] n=4 b a f ( x0 ) 2 f ( x1 ) 2 f ( x2 ) ... f ( xn ) 2n 1 1 1 3 T f (0) 2 f ( ) 2 f ( ) 2 f ( ) f (1)

8 4 2 4 1 1 1 9 11 2 T 0 1 units 8 8 2 8 32 Just like in the Riemann Sums, if the area can be found by hand, you can sue that answer to check to see how close the estimate was to the exact answer. In this particular problem, the exact answer is 1/3units squared or .3333 units squared. Using the Trapezoidal Rule, the estimate comes out to be .34375 units squared. The estimated answer is very close to the exact answer. TRY THIS PROBLEM:

1 Calculate the Trapezoidal Rule for f ( x) x on the interval [1,2] when n=5 x0 1 b a 1 n 5 b a 1 2n 10 6 x1 5 7 x2 5 8 x3

5 9 x4 5 x5 2 ANSWER 1 5 5 5 5 1 1753 T 1 2( ) 2( ) 2( ) 2( ) .6956units 2 10 6 7 8 9 2 2520 SIMPSONS RULE Simpsons Rule is more accurate than both the Riemann Sums and the Trapezoidal Rule. The

Simpsons Rule uses various figures to fill in the area under a curve in order to estimate the area The formula for the Simpsons Rule is: b a f ( x0 ) 4 f ( x1 ) 2 f ( x2 ) 4 f ( x3 ).... f ( xn ) 3n *When using the Simpsons Rule n can NOT be an odd number b a n Is still added to each xnumber just like in the Riemann Sum and in the Trapezoidal Rule EXAMPLE PROBLEM Calculate the Simpsons Rule for f ( x) x

on the interval [0,4] using n=4 b a 1 n x0 0 x1 1 x2 2 x3 3 x4 4 b a 1 3n 3 Now plug all the information found into the formula b a

f ( x0 ) 4 f ( x1 ) 2 f ( x2 ) 4 f ( x3 ).... f ( xn ) 3n 1 S f (0) 4 f (1) 2 f (2) 4 f (3) f (4) 3 1 2 S [o 4 2 2 4 3 2] 5.2522units 3 TRY THIS PROBLEM: Calculate the Simpsons Rule for f ( x) 9 x on [3,5] using n=4 b a 1 n 2

x0 3 7 x1 2 x2 4 9 x3 2 x4 5 b a 1 3n 6 2 ANSWER 1 7 9 S f (3) 4 f ( ) 2 f (4) 4 f ( ) f (5)

6 2 2 1 44 2 S 0 13 14 45 16 units 6 3 CLASSWORK 1. Find the Left Riemann of f ( x ) x 3 on [0,2] when n=6

3 2. Find the Right Riemann of f ( x) x on [0,2] when n=6 3. Find the Midpoint Riemann of f ( x) x3 on [0,2] when n=6 4. Calculate the Trapezoidal rule for f ( x) sin x on [0,] using 4 rectangles.] for n=4 5. Calculate the Simpsons Rule on [2,4] where n=4 1 for 1 x2 CLASSWORK ANSWERS 1. L 1 0 1 8 1 64 125 76 units 2 R 3

27 27 27 27 27 2. RR 1 8 units 2 3 27 27 27 27 9 1 1 64 125 49 1 1 1 125 343 27 1331 2753

2 3. M R 1 units 3 216 27 216 216 8 216 648 4. T 0 2 8 8 2 22 2

2 0 2.1063units 2 2 11 4 1 4 1 2 S 4( ) 4( )

.2187 units 5. 6 5 29 5 53 17 BIBLIOGRAPHY An Approximation of the integral of f(x)=x^2 on the interval [0, 100] Using a Midpoint Riemann Sum. Riemann. Web. 6 Mar. 2011. Anton, Howard. Calculus A New Horizon. Sixth ed. New York: John Wiley & Sons, Inc., 1999. Print. Bartkovich, Kevin, John Goebel, Julie Graves, and Daniel Teague. Contemporary Calculus through applications. Chicago, Illinois: Everyday Learning Corporation, 1999. Print. Beeson, Michael. It is possible to make a Riemann Sum. Riemann Sums, San Jose, California. MathXpert: Learning Mathematics in the 21st Century. Web. 6 Mar. 2011. Karl. Right Riemann Sum of a Parabola. Section 10: Integrals, Karl's Calc Tutor. http://www.karlscalculus.org/calc10_0.html. Web. 6 Mar. 2011. "ListenToYouTube.com: Youtube to MP3, get mp3 from youtube video, flv to mp3, extract audio from youtube, youtube mp3." Convert YouTube to MP3, Get MP3 from YouTube video, FLV to MP3, extract audio from YouTube,

YouTube MP3 - ListenToYouTube.com.. Web. 6 Mar. 2011.

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