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.

Introduction to area under a curve (cont.)

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

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

  • 6 partitions
  • Volunteer:_________

!Show Your Work!

  • n=6

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

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!

  • 4 partitions
  • Volunteer:____________

!Show Your Work!

  • n=4

Check Your Answer!

Sources

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


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