Section 4.3 – Riemann Sums and Definite Integrals

Section 4.3 – Riemann Sums and Definite Integrals

Riemann Sums The rectangles need not have equal width, and the height may be any value of f(x) within the subinterval. 1. Partition (divide) [a,b] into N subintervals. 2. Find the length of each interval: 3. Find any point ci in the interval [xi,xi-1]. ci cN c2 c1 c3 x2 a =x0 x3 x1 xi =xN 4. Construct every rectangle of height f(ci) and base Δxi. b 4. Find the sum of the areas.

Riemann Sums The norm of P, denoted ││P││, is the maximum of the lengths Δxi. a b As ││P││ gets closer to 0, the sum of the areas of the rectangles is closer to the actual area under the curve

Definite Integral The definite integral of f(x) over [a,b], denoted by the integral sign, is the limit of Riemann sums: Where the limit exists, we say that f(x) is integrable over [a,b]. Upper limit of integration Lower limit of integration

Notation Examples The definite integral that represents the area is… EX1: f(x) S a b Ex2: The area under the parabola y=x2 from 0 to 1

Theorem: The Existence of Definite Integrals If f(x)is continuous on [a,b], or if f(x) is continuous with at most finitely many jump discontinuities (one sided limits are finite but not equal), then f(x) is integrable over [a,b]. a b

Negative Area or “Signed” Area If a function is less than zero for an interval, the region between the graph and the x-axis represents negative area. Positive Area Negative Area

Definite Integral: Area Under a Curve If y=f(x) is integrable over a closed interval [a,b], then the area under the curve y=f(x) from a to b is the integral of f from a to b. Upper limit of integration Lower limit of integration

Example 1 Calculate .

Example 2 Calculate .

Rules for Definite Integrals Let f and g be functions and x a variable; a, b, c, and k be constant.

Example 1 If , calculate . Sum Rule Constant Multiple Rule Given and Constant Rule

Example 2 If , calculate . Additivity Rule Given

Section 4.3 – Riemann Sums and Definite Integrals