Exploring Reiman Sum with GeoGebra
Example
Riemann sum is a method used to approximate the area under a curve by dividing the region into smaller rectangles and summing their individual areas. It is an integral approximation technique introduced by the mathematician Bernhard Riemann.
Mathematical Formulas:
Suppose we have a function f(x) defined on the interval [a, b] and we want to approximate the area under the curve using Riemann sum. There are different types of Riemann sums, but two common ones are left Riemann sum and right Riemann sum.
1. Left Riemann Sum:
The left Riemann sum uses the left endpoint of each subinterval to calculate the height of the rectangle. The formula for the left Riemann sum is:
L(f, P) = ∑[i=0 to n-1] f(x_i) * Δx
where:
- f(x_i) represents the value of the function at the left endpoint of each subinterval,
- Δx is the width of each subinterval,
- n is the number of subintervals, and
- P is the partition of the interval [a, b] into subintervals.
2. Right Riemann Sum:
The right Riemann sum uses the right endpoint of each subinterval to calculate the height of the rectangle. The formula for the right Riemann sum is:
R(f, P) = ∑[i=1 to n] f(x_i) * Δx
where:
- f(x_i) represents the value of the function at the right endpoint of each subinterval,
- Δx is the width of each subinterval,
- n is the number of subintervals, and
- P is the partition of the interval [a, b] into subintervals.
Examples:
1. Let's approximate the area under the curve y = x^2 on the interval [0, 1] using a left Riemann sum with 4 subintervals.
Using a partition P = {0, 0.25, 0.5, 0.75, 1}, where Δx = (1-0)/4 = 0.25:
L(f, P) = f(0) * 0.25 + f(0.25) * 0.25 + f(0.5) * 0.25 + f(0.75) * 0.25
= 0 * 0.25 + 0.0625 * 0.25 + 0.25 * 0.25 + 0.5625 * 0.25
= 0.15625
The left Riemann sum approximation of the area under the curve is 0.15625.
2. Consider the function f(x) = 2x + 1 on the interval [1, 4]. Let's use a right Riemann sum with 3 subintervals to approximate the area under the curve.
Using a partition P = {1, 2.5, 4}, where Δx = (4-1)/3 = 1:
R(f, P) = f(1) * 1 + f(2.5) * 1 + f(4) * 1
= (2*1 + 1) * 1 + (2*2.5 + 1) * 1 + (2*4 + 1) * 1
= 1 + 6.5 + 9
= 16.5
The right Riemann sum approximation of the area under the curve is 16.5.
These examples demonstrate the use of Riemann sum to approximate the area under a curve by dividing the region into smaller rectangles and summing their areas. The choice of left or right Riemann sum depends on the endpoints of the subintervals used in the approximation.