# Finding the area under a curve: Riemann, Trapezoidal, and Simpson’s Rule

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## 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.

## 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 TRY ME!

• Volunteer:___________________
• 4 Partitions

• n=4

• 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- TRY ME!

• Volunteer:___________
• 3 Partitions

• n=3

• 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- TRY ME

• 6 partitions
• Volunteer:_________

• n=6

• 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- TRY Me

• Volunteer:_____________
• 4 Partitions

• n=4

• 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 TRY ME!

• 4 partitions
• Volunteer:____________

• n=4

## Sources

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

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