9.2 Properties of Riemann Integrable Functions

Riemann integrals are a key concept in calculus, allowing us to calculate areas under curves. They have special properties that make them super useful in math and science.

These properties include linearity, additivity, and comparison. Understanding them helps us solve complex problems and connect different areas of math. Let's dive into what makes Riemann integrals so powerful!

Linearity of Riemann Integrals

Definition and Proof

Top images from around the web for Definition and Proof

• 

  Integral - Wikipedia View original

  Is this image relevant?

• 

  Riemann integral - Wikipedia View original

  Is this image relevant?

• 

  real analysis - Understanding definition of Riemann Integral - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Integral - Wikipedia View original

  Is this image relevant?

• 

  Riemann integral - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Definition and Proof

• 

  Integral - Wikipedia View original

  Is this image relevant?

• 

  Riemann integral - Wikipedia View original

  Is this image relevant?

• 

  real analysis - Understanding definition of Riemann Integral - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Integral - Wikipedia View original

  Is this image relevant?

• 

  Riemann integral - Wikipedia View original

  Is this image relevant?

1 of 3

• The linearity property states that for Riemann integrable functions $f$ and $g$ on $[a,b]$ and constants $c$ and $d$, the integral of $cf + dg$ over $[a,b]$ equals $c$ times the integral of $f$ plus $d$ times the integral of $g$
  • In mathematical notation: $\int_a^b (cf(x) + dg(x)) dx = c \int_a^b [f(x)](https://www.fiveableKeyTerm:f(x)) dx + d \int_a^b g(x) dx$
• To prove the linearity property, consider the Riemann sums for $f$ and $g$ separately, then combine them using the properties of limits and the definition of the Riemann integral
  • Let $P$ be a partition of $[a,b]$ and $x_i^*$ be a sample point in the $i$-th subinterval of $P$
  • The Riemann sum for $cf + dg$ is $\sum_{i=1}^n (cf(x_i^*) + dg(x_i^*)) \Delta x_i = c \sum_{i=1}^n f(x_i^*) \Delta x_i + d \sum_{i=1}^n g(x_i^*) \Delta x_i$
  • As the norm of $P$ approaches zero, the Riemann sums converge to the integrals, proving the linearity property

Applications and Extensions

• The linearity property allows for the integration of linear combinations of Riemann integrable functions, simplifying the calculation of integrals in many cases
  • Example: $\int_0^1 (3x^2 + 2e^x) dx = 3 \int_0^1 x^2 dx + 2 \int_0^1 e^x dx$
• The linearity property can be extended to finite sums of Riemann integrable functions multiplied by constants
  • For functions $f_1, f_2, \ldots, f_n$ and constants $c_1, c_2, \ldots, c_n$: $\int_a^b (\sum_{i=1}^n c_i f_i(x)) dx = \sum_{i=1}^n c_i \int_a^b f_i(x) dx$
• The linearity property is a fundamental tool in the study of differential equations, Fourier analysis, and many other areas of mathematics and physics

Additivity of Riemann Integrals

Additivity over Subintervals

• The additivity property states that if $f$ is Riemann integrable on $[a,b]$ and $c$ is any point in $(a,b)$, then $f$ is Riemann integrable on $[a,c]$ and $[c,b]$, and the integral of $f$ over $[a,b]$ equals the sum of the integrals of $f$ over $[a,c]$ and $[c,b]$
  • In mathematical notation: $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$
• The additivity property allows for the division of the interval of integration into subintervals, which can simplify the calculation of integrals or enable the application of different techniques on each subinterval
  • Example: To integrate $f(x) = \begin{cases} x^2, & x \in [0,1] \\ e^x, & x \in (1,2] \end{cases}$, use additivity: $\int_0^2 f(x) dx = \int_0^1 x^2 dx + \int_1^2 e^x dx$

Extensions and Consequences

• The additivity property can be extended to any finite number of subintervals, provided that the function is Riemann integrable on each subinterval
  • For a partition $a = x_0 < x_1 < \ldots < x_n = b$: $\int_a^b f(x) dx = \sum_{i=1}^n \int_{x_{i-1}}^{x_i} f(x) dx$
• The additivity property is a consequence of the linearity property and the definition of the Riemann integral as a limit of Riemann sums
  • Proof: Apply the linearity property to the sum of the integrals over the subintervals and use the uniqueness of the Riemann integral
• The additivity property is essential in the development of the fundamental theorem of calculus and the study of improper integrals

Comparison Property for Integrability

Statement and Consequences

• The comparison property states that if $f$ and $g$ are Riemann integrable functions on $[a,b]$ with $f(x) \leq g(x)$ for all $x$ in $[a,b]$, then the integral of $f$ over $[a,b]$ is less than or equal to the integral of $g$ over $[a,b]$
  • In mathematical notation: If $f(x) \leq g(x)$ for all $x \in [a,b]$, then $\int_a^b f(x) dx \leq \int_a^b g(x) dx$
• The comparison property allows for the estimation of integrals by comparing the integrand to simpler functions with known integrals
  • Example: To estimate $\int_0^1 \sin(x^2) dx$, use the inequality $0 \leq \sin(x^2) \leq x^2$ for $x \in [0,1]$, so $0 \leq \int_0^1 \sin(x^2) dx \leq \int_0^1 x^2 dx = \frac{1}{3}$
• The comparison property can be used to establish bounds for the value of an integral, which is useful in approximation and error analysis
  • Example: If $|f(x) - g(x)| \leq \varepsilon$ for all $x \in [a,b]$, then $|\int_a^b f(x) dx - \int_a^b g(x) dx| \leq \varepsilon(b-a)$

Strict Inequality and Almost Everywhere

• The strict inequality holds in the comparison property if $f(x) < g(x)$ for at least one $x$ in $[a,b]$ and $f$ is not equal to $g$ almost everywhere on $[a,b]$
  • Almost everywhere means that the set of points where $f(x) = g(x)$ has measure zero
• If $f(x) \leq g(x)$ for all $x \in [a,b]$ and $f(x) = g(x)$ almost everywhere on $[a,b]$, then $\int_a^b f(x) dx = \int_a^b g(x) dx$
  • This is a consequence of the uniqueness of the Riemann integral for a given function
• The comparison property and its strict inequality variant are crucial in the study of optimization problems and the convergence of sequences and series of functions

Integrability of Functions

Sufficient Conditions for Integrability

• A function $f$ is Riemann integrable on $[a,b]$ if and only if it is bounded on $[a,b]$ and continuous almost everywhere on $[a,b]$
  • Boundedness ensures that the upper and lower Riemann sums are finite
  • Continuity almost everywhere guarantees the existence of the limit of Riemann sums
• Discontinuities of a function can be classified as removable, jump, or infinite discontinuities. A function with a finite number of jump discontinuities is Riemann integrable
  • Example: The function $f(x) = \begin{cases} 1, & x = 0 \\ 0, & x \neq 0 \end{cases}$ has a jump discontinuity at $x = 0$ but is Riemann integrable on any interval
• Piecewise-defined functions are Riemann integrable if each piece is Riemann integrable and the function has only a finite number of discontinuities
  • Example: The absolute value function $f(x) = |x|$ is Riemann integrable on any interval, as it is piecewise-defined by two continuous functions with a single discontinuity at $x = 0$

Preservation of Integrability

• The sum, difference, product, and quotient of Riemann integrable functions are also Riemann integrable, provided that the quotient function's denominator is non-zero almost everywhere on the interval
  • Example: If $f$ and $g$ are Riemann integrable on $[a,b]$ and $g(x) \neq 0$ for all $x \in [a,b]$, then $\frac{f}{g}$ is Riemann integrable on $[a,b]$
• The composition of a Riemann integrable function with a continuous function is Riemann integrable
  • Example: If $f$ is Riemann integrable on $[a,b]$ and $g$ is continuous on the range of $f$, then $g \circ f$ is Riemann integrable on $[a,b]$
• The absolute value, maximum, and minimum of Riemann integrable functions are also Riemann integrable
  • Example: If $f$ and $g$ are Riemann integrable on $[a,b]$, then $\max\{f,g\}$ and $\min\{f,g\}$ are Riemann integrable on $[a,b]$