Adguary Calwile Laura Rogers Autrey~ 2nd Per. 3/14/11 Finding the area under a curve: Riemann, Trapezoidal, and Simpson’s Rule Adguary Calwile Laura Rogers Autrey~ 2nd Per. 3/14/11

Introduction to area under a curve Before integration was developed, people found the area under curves by dividing the space beneath into rectangles, adding the area, and approximating the answer. As the number of rectangles, n, increases, so does the accuracy of the area approximation.

Introduction to area under a curve (cont.) There are three methods we can use to find the area under a curve: Riemann sums, the trapezoidal rule, and Simpson’s rule. For each method we must know: f(x)- the function of the curve n- the number of partitions or rectangles (a, b)- the boundaries on the x-axis between which we are finding the area

Finding Area with Riemann Sums Subintervals with equal width For convenience, the area of a partition is often divided into subintervals with equal width – in other words, the rectangles all have the same width. (see the diagram to the right for an example of a Right Riemann approximation)

Finding Area with Riemann Sums It is possible to divide a region into different sized rectangles based on an algorithm or rule (see graph above)

Riemann Sums There are three types of Riemann Sums Right Riemann: Left Riemann: Midpoint Riemann:

Right Riemann- Overview Right Riemann places the right point of the rectangles along the curve to find the area. The equation that is used for the RIGHT RIEMANN ALWAYS begins with: And ends with Within the brackets!

Right Riemann- Example Remember: Right Only Given this problem below, what all do we need to know in order to find the area under the curve using Right Riemann? 4 partitions

Right Riemann- Example For each method we must know: f(x)- the function of the curve n- the number of partitions or rectangles (a, b)- the boundaries on the x-axis between which we are finding the area

Right Riemann- Example

Right Riemann TRY ME! Volunteer:___________________ 4 Partitions

!Show All Your Work! n=4

Did You Get It Right? n=4

Left Riemann- Overview Left Riemann uses the left corners of rectangles and places them along the curve to find the area. The equation that is used for the LEFT RIEMANN ALWAYS begins with: And ends with Within the brackets!

Left Riemann- Example Remember: Left Only Given this problem below, what all do we need to know in order to find the area under the curve using Left Riemann? 4 partitions

Left Riemann- Example For each method we must know: f(x)- the function of the curve n- the number of partitions or rectangles (a, b)- the boundaries on the x-axis between which we are finding the area

Left Riemann- Example

Volunteer:___________ Left Riemann- TRY ME! Volunteer:___________ 3 Partitions

!Show All Your Work! n=3

Did You Get My Answer? n=3

Midpoint Riemann- Overview Midpoint Riemann uses the midpoint of the rectangles and places them along the curve to find the area. The equation that is used for MIDPOINT RIEMANN ALWAYS begins with: And ends with Within the brackets!

Midpoint Riemann- Example Remember: Midpoint Only Given this problem below, what all do we need to know in order to find the area under the curve using Midpoint Riemann? 4 partitions

Midpoint Riemann- Example For each method we must know: f(x)- the function of the curve n- the number of partitions or rectangles (a, b)- the boundaries on the x-axis between which we are finding the area

Midpoint Riemann- Example

Midpoint Riemann- TRY ME Volunteer:_________ 6 partitions

!Show Your Work! n=6

Correct??? n=6

Applications of Approximating Areas EXAMPLE The velocity of a car (in feet per second) is recorded from the speedometer every 10 seconds, beginning 5 seconds after the car starts to move. See Table 2. Use a Riemann sum to estimate the distance the car travels during the first 60 seconds. (Note: Each velocity is given at the middle of a 10-second interval. The first interval extends from 0 to 10, and so on.) SOLUTION Since measurements of the car’s velocity were taken every ten seconds, we will use . Now, upon seeing the graph of the car’s velocity, we can construct a Riemann sum to estimate how far the car traveled. Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #26

Applications of Approximating Areas This is an example of using a midpoint Riemann sum to approximate an integral. Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #27

Applications of Approximating Areas CONTINUED Therefore, we estimate that the distance the car traveled is 2800 feet. Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #28

Trapezoidal Rule Overview Trapezoidal Rule is a little more accurate than Riemann Sums because it uses trapezoids instead of rectangles. You have to know the same 3 things as Riemann but the equation that is used for TRAPEZOIDAL RULE ALWAYS begins with: and ends with Within the brackets with every“ f ” being multiplied by 2 EXCEPT for the first and last terms

Trapezoidal Rule- Example Remember: Trapezoidal Rule Only Given this problem below, what all do we need to know in order to find the area under the curve using Trapezoidal Rule? 4 partitions

Trapezoidal Example For each method we must know: f(x)- the function of the curve n- the number of partitions or rectangles (a, b)- the boundaries on the x-axis between which we are finding the area

Trapezoidal Rule- Example

Trapezoidal Rule- TRY Me Volunteer:_____________ 4 Partitions

Trapezoidal Rule- TRY ME!! n=4

Was this your answer? n=4

Simpson’s Rule- Overview Simpson’s rule is the most accurate method of finding the area under a curve. It is better than the trapezoidal rule because instead of using straight lines to model the curve, it uses parabolic arches to approximate each part of the curve. The equation that is used for Simpson’s Rule ALWAYS begins with: And ends with Within the brackets with every “f” being multiplied by alternating coefficients of 4 and 2 EXCEPT the first and last terms. In Simpson’s Rule, n MUST be even.

Simpson’s Rule- Example Remember: Simpson’s Rule Only Given this problem below, what all do we need to know in order to find the area under the curve using Simpson’s Rule? 4 Partitions

Simpson’s Example For each method we must know: f(x)- the function of the curve n- the number of partitions or rectangles (a, b)- the boundaries on the x-axis between which we are finding the area

Simpson’s Rule- Example

Simpson’s Rule TRY ME! Volunteer:____________ 4 partitions

!Show Your Work! n=4

Check Your Answer!

Sources http://www.intmath.com/Integration © Laura Rogers, Adguary Calwile; 2011