Area Under a Curve

The applet below shows the relationship among the area under the curve, the lower sum, and the upper sum. As the number of rectangles n increases, the sum of the areas of the rectangles get closer and closer to the actual area under the curve.
The approximation above is called the Riemann Sum. It was named after Bernhard Riemann, a German mathematician. 

Angle Sum Theorem

Move points A, B, and C to choose the triangle size and shape you want.  Move sliders and to the extreme right.

  1. What do you observe about angles 1, 2 and 3
  2. What is the sum of angles 1, 2, and 3? Justify your answer. 
  3. Based on your observations above, what can you say about the sum of the interior angles of a triangle? Explain your answer. 
 
 

Approximating Pi

Move slider n to the right and observe what happens to the inscribed polygon.
 
The applet above shows the relationship between the area of a circle and the inscribed polygon.  Notice that as the the number of sides (n) of the inscribed polygon increases, its area gets closer and closer to the area of the circle which is equal to .  This approximation is called the method of exhaustion. It was developed used by Archimedes.

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