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 [f(x)](https://www.fiveableKeyTerm:f(x)) on a closed [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 exists for f(x) on [a,b], then f(x) is on [a,b]
Riemann Sums and Integrability
The Riemann integral is defined as the limit of Riemann sums as the norm of the 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 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 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
Key Terms to Review (19)
∫: The symbol ∫ represents the integral in mathematics, which is a fundamental concept for calculating the area under curves or the accumulation of quantities. Integrals can be defined in various ways, with Riemann integrals focusing on partitioning intervals and summing up areas of rectangles, while also playing a crucial role in connecting derivatives and integration through the Fundamental Theorem of Calculus.
Augustin-Louis Cauchy: Augustin-Louis Cauchy was a French mathematician whose work laid the groundwork for modern analysis, particularly in the study of limits, continuity, and integrals. His contributions, including the formalization of the concept of a limit and the development of the Riemann integral, have had a profound impact on mathematical analysis and are foundational to various important results and theorems.
Bernhard Riemann: Bernhard Riemann was a German mathematician whose work laid the foundations for several important areas in mathematics, particularly in analysis and geometry. He is best known for his contributions to the concept of integration, which is crucial for understanding how to calculate areas under curves and the behavior of functions. His ideas extend to the convergence of sequences and series, providing essential tools for studying continuity and differentiability.
Bounded function: A bounded function is a function whose values stay within a fixed range, meaning there exist real numbers, say $m$ and $M$, such that for all inputs $x$ in the domain, the output satisfies $m \leq f(x) \leq M$. This property of boundedness is crucial in various mathematical concepts, as it ensures that the function does not diverge or become infinite, making it essential for understanding integrability, continuity, and optimization.
Continuous Function: A continuous function is a type of function where small changes in the input result in small changes in the output. This means that as you approach a certain point on the function, the values of the function get closer and closer to the value at that point. This concept connects deeply with various mathematical ideas, such as integrability, differentiation, and limits, shaping many fundamental theorems and properties in calculus.
F(x): In mathematical analysis, f(x) represents a function where 'f' is the name of the function and 'x' is the input variable. Functions like f(x) map inputs to outputs, and they can exhibit various properties, such as continuity, differentiability, and integrability. Understanding f(x) is crucial when studying the behavior of functions in relation to limits, approximations, and series expansions.
Fundamental theorem of calculus: The fundamental theorem of calculus establishes a deep connection between differentiation and integration, showing that these two operations are essentially inverse processes. It consists of two parts: the first part guarantees that if a function is continuous on an interval, then it has an antiderivative, while the second part provides a method to evaluate definite integrals using antiderivatives. This theorem is pivotal in understanding how integration can be applied to calculate areas and solve real-world problems.
Interval: An interval is a set of real numbers that contains all numbers between any two numbers in the set. It can be open, closed, or half-open, which influences how we handle limits and continuity in analysis. Understanding intervals is essential for defining the behavior of functions over specific ranges and plays a critical role in concepts like integrals, derivatives, and the continuity of functions.
Limit of Riemann Sums: The limit of Riemann sums is the value that Riemann sums approach as the partition of the interval becomes infinitely fine, effectively capturing the area under a curve. This concept is crucial for defining the Riemann integral, which represents the total accumulation of quantities over an interval. As the number of subintervals increases and their widths decrease, these sums converge to a single value, which is interpreted as the area under the curve represented by a function on that interval.
Lower Sum: The lower sum is a method used to approximate the area under a curve by partitioning the interval into smaller subintervals and taking the minimum value of the function on each subinterval. This technique is essential in understanding the Riemann integral, as it helps establish a way to estimate the total area by summing these minimum values multiplied by the width of the subintervals. The lower sum is key to exploring the properties of Riemann integrable functions, allowing for comparisons with upper sums to determine integrability.
Partition: In mathematical analysis, a partition is a division of an interval into smaller subintervals, which helps in approximating the area under a curve. This concept is crucial for defining the Riemann integral as it establishes how the interval is broken down to calculate Riemann sums, which serve as approximations of the integral. The choice of partition directly affects the accuracy of these approximations and highlights properties of integrable functions.
Piecewise Function: A piecewise function is a mathematical function defined by multiple sub-functions, each of which applies to a specific interval or condition. This allows for different expressions to be used for different parts of the domain, making it useful in modeling situations where behavior changes based on certain thresholds or values. Piecewise functions can help in understanding continuity, integration, and differentiability within various mathematical contexts.
Pointwise Convergence: Pointwise convergence refers to a type of convergence of a sequence of functions where, for each point in the domain, the sequence converges to the value of a limiting function. This means that for every point, as you progress through the sequence, the values get closer and closer to the value defined by the limiting function. Pointwise convergence is crucial in understanding how functions behave under limits and is often contrasted with uniform convergence, which has different implications for continuity and integration.
Properties of Integrals: Properties of integrals are fundamental rules that describe how integrals behave under various operations, providing a framework for simplifying and solving integral expressions. These properties include linearity, additivity, and the ability to interchange limits and integration, which are essential when working with Riemann integrals. Understanding these properties helps in effectively calculating areas under curves and solving problems involving definite and indefinite integrals.
Riemann Integrable: A function is called Riemann integrable if it can be approximated by Riemann sums over its interval, and if the limit of these sums exists as the partition of the interval becomes finer. This concept is central to understanding how to calculate the area under a curve using limits and partitions, emphasizing that a Riemann integrable function must be bounded and its set of discontinuities must have measure zero.
Riemann Sum: A Riemann sum is a method for approximating the total area under a curve by dividing the region into smaller subintervals, calculating the area of rectangles based on function values at certain points within those intervals, and summing these areas. This concept is foundational in calculus and helps establish the formal definition of the Riemann integral, which allows for determining the exact area under a curve as the number of subdivisions approaches infinity.
Squeeze Theorem: The Squeeze Theorem is a mathematical principle that helps find the limit of a function by comparing it to two other functions that 'squeeze' it. When one function approaches a limit from above and another from below, and both converge to the same value, the function in between must also approach that value. This concept is crucial for establishing limits in various contexts, including sequences and functions.
Upper Sum: An upper sum is a method used to approximate the area under a curve by summing the areas of rectangles that lie above the graph of a function. This technique is crucial in the study of Riemann integrals, as it helps in understanding how to estimate the integral value and establishes bounds for the actual area under the curve.
δx: The symbol δx represents a small change or increment in the variable x, often used in the context of Riemann integrals and sums. It is critical for understanding how to partition the interval of integration into smaller segments, which is essential for approximating the area under a curve. This concept connects with various properties of integrable functions and helps in determining the total area through summation techniques.