Understanding Riemann Sums and Definite Integrals in Calculus
In the 1800s, mathematician Georg Riemann pioneered the concept of Riemann Sums to define the area under a curve. By partitioning a region into rectangles, where the height of each rectangle is determined by the function value at specified points, the limit of these sums leads to the definition of a definite integral. If a function f is integrable on the interval [a, b], the definite integral can be calculated, representing the area under the curve. This area is given by the integral symbol ∫ and is bounded by lower limit a and upper limit b, providing essential insights in calculus.
Understanding Riemann Sums and Definite Integrals in Calculus
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Presentation Transcript
Section 4.3 • In the 1800’s, the German mathematician, Georg Riemann, used the limit of a sum to define the area of a region in a plane.
Riemann Sum n = # of rectangles (partitions) f (xi) = height of each rectangle ∆xi = width of each rectangle
Consider the following limit: = L → Area under the curve
Definition of a Definite Integral • If f is defined on the closed interval [a, b] and • exists, then f is integrable (can be integrated) on [a, b] and the limit is denoted by This symbol means the sum from a to b.
The limit is called the definite integral. This is always a number. • The number “a” is called the lower limit of integration. • The number “b” is called the upper limit of integration. • The function “f (x)” is called the integrand.
Area under a curve can be represented using a definite integral. f (x)
8 Area of rectangle = L ∙ W = 2 ∙ 4
4 Area of rt. ∆
2 Area of semicircle
-3 Area I is a negative #. II Area II is a positive #. I
0 No area under the curve
Split the interval into parts like ex. 4.
HW: p. 278 (13-43 odd, 46, 47, 49)
