# Adguary Calwile Laura Rogers Autrey~ 2nd Per. 3/14/11

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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

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

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

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

Correct??? n=6

Trapezoidal Rule Overview
Trapezoidal Rule is a little more accurate that 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

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