The Riemann integral is a fundamental concept in calculus, defining how we measure the area under a curve. It's all about breaking up a function into tiny pieces, adding them up, and seeing what happens as those pieces get infinitely small.
This idea connects to the broader study of integration by providing a rigorous way to calculate areas and volumes. Understanding the Riemann integral helps us tackle more complex integration problems and lays the groundwork for advanced calculus concepts.
Riemann Integral for Bounded Functions
Definition and Notation
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The Riemann integral defines the integral of a bounded function f(x) on a closed interval [a,b]
f(x) is bounded on [a,b] if there exist real numbers m and M such that m≤f(x)≤M for all x in [a,b]
The Riemann integral of f(x) over [a,b] is denoted as ∫abf(x)dx
If the limit of Riemann sums exists for f(x) on [a,b], then f(x) is Riemann integrable on [a,b]
Riemann Sums and Integrability
The Riemann integral is defined as the limit of Riemann sums as the norm of the partition approaches zero, if this limit exists
The norm (or mesh) of a partition P, denoted as ∣∣P∣∣, is the length of the largest subinterval in the partition
The Riemann sum approximates the area under the curve of f(x) over [a,b] by summing the areas of rectangles with heights f(ξi) and widths Δxi
A function f(x) is Riemann integrable on [a,b] if and only if for every ε>0, there exists a δ>0 such that for any partition P with ∣∣P∣∣<δ and any choice of sample points, ∣S(P,f)−I∣<ε, where I is the value of the integral
Partitions and Riemann Sums
Partitions and Subintervals
A partition P of a closed interval [a,b] is a finite set of points {x0,x1,…,xn} such that a=x0<x1<…<xn=b
The subintervals [xi−1,xi] for i=1,2,…,n are called the subintervals of the partition P
For example, if P={0,1,2,3} on the interval [0,3], the subintervals are [0,1], [1,2], and [2,3]
Calculating Riemann Sums
For each subinterval [xi−1,xi], choose a sample point ξi in [xi−1,xi]
The Riemann sum of f(x) with respect to the partition P and the chosen sample points is defined as S(P,f)=∑i=1nf(ξi)Δxi, where Δxi=xi−xi−1
For example, if f(x)=x2 on [0,1] with partition P={0,0.5,1} and sample points ξ1=0.25 and ξ2=0.75, then S(P,f)=f(0.25)⋅0.5+f(0.75)⋅0.5=0.0625⋅0.5+0.5625⋅0.5=0.3125
Integrability of Functions
Conditions for Integrability
Continuous functions on a closed interval [a,b] are always Riemann integrable
For example, f(x)=sin(x) is continuous on any closed interval and thus Riemann integrable
Monotonic functions on a closed interval [a,b] are always Riemann integrable
For example, f(x)=x3 is monotonically increasing on any closed interval and thus Riemann integrable
Bounded functions with a finite number of discontinuities on a closed interval [a,b] are Riemann integrable
For example, f(x)={1,0,x∈Qx∈/Q is bounded and has infinitely many discontinuities on any interval, so it is not Riemann integrable
Upper and Lower Riemann Integrals
f(x) is Riemann integrable on [a,b] if and only if the upper and lower Riemann integrals of f(x) over [a,b] are equal
The upper Riemann integral is the infimum of the upper Riemann sums, while the lower Riemann integral is the supremum of the lower Riemann sums
To compute a Riemann integral using the definition, find the limit of Riemann sums as the norm of the partition approaches zero
For example, to compute ∫01x2dx, consider the partition Pn={0,n1,n2,…,1} and the sample points ξi=ni. The Riemann sum is S(Pn,f)=∑i=1n(ni)2n1=n31∑i=1ni2. As n→∞, this sum approaches 31, which is the value of the integral
Properties of Riemann Integrals
Linearity: For constants a and b, ∫ab[af(x)+bg(x)]dx=a∫abf(x)dx+b∫abg(x)dx
Additivity: If c is in [a,b], then ∫abf(x)dx=∫acf(x)dx+∫cbf(x)dx
Monotonicity: If f(x)≤g(x) for all x in [a,b], then ∫abf(x)dx≤∫abg(x)dx
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus relates the Riemann integral to the antiderivative of a function, providing a powerful tool for computing integrals
If F(x) is an antiderivative of f(x) on [a,b], then ∫abf(x)dx=F(b)−F(a)
For example, since F(x)=31x3 is an antiderivative of f(x)=x2, we have ∫01x2dx=F(1)−F(0)=31−0=31