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RiemannSums

Page history last edited by LFS 11 years, 5 months ago

Home -> Do Mathematics -> Calculus -> Integration -> Riemann Sums


Riemann Sums - Classroom/Home Simulator  (See also: Students Building Simulators - Riemann Sums)

 

GeoGebra InterActivity Directions for InterActivity   Formulas + Animation 

 

1. To change the number of intervals, click and drag slider n.

2. To change interval of integration, type new values for a and b on the input line: example: a=0 (enter) and then b = 10 (enter).

3. To change the function, type new f(x) on input line.

4. Use ctrl+mouse scroll to zoom and shift +mouse drag to pan (or double-click to open in geogebra).

5. To turn off the Algebra view (at left), click on View -> Algebra View.

 
 

 

Formula       Formula  

   Formula

 

 

 

 
 

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

 

Some Questions

  • Q1. A Riemann Sum should get closer and closer to the definite integral as the number n of subintervals gets bigger and bigger. Define LeftErr=abs(LeftSum-c). What is LeftErr when n=2, when n=5 and when n=10? Is it decreasing?
  • Q2. Define f(x)=3, a=1, b=5. Geometrically, what is the shape of the area under the curve in the interval [1,5]? Find this area A using a standard formula for area. What is the value of the definite integral ? Set n=1. What is the Left Riemann Sum? Why is it the same as the definite integral? Set n=2, then n=5 and then n=10. Why doesn’t the Left Riemann Sum change?
  • Q3. Define f(x)=-3 (a=1, b=5). Geometrically, what is the shape of the area under the curve in the interval [1,5]? Find this area A using a standard formula for area. What is the value of the definite integral ? Why is A different than I? Set n=1. What is the Left Riemann Sum? Why is it the same as the definite integral? Set n=2, then n=5 and then n=10. Why doesn’t the Left Riemann Sum change?
  • Q4. Define f(x)=sin(x), a=0, b=2*pi. What is the value of the definite integral ? What is LeftErr when n=2, when n=5 and when n=10? Why is the LeftSum so “accurate”?
  • Q5. Change b=pi/2. What is the value of the definite integral ? What is LeftErr when n=2, when n=5 and when n=10? Is LeftSum so “accurate” now?

 

 

Home -> Do Mathematics -> Calculus-> Integration -> Riemann Sums

calculus, functions, riemann, sum, integration, geogebra, application, geometry, program, applet, simulator

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