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−  There are a few equivalent definitions of $\Delta$integration.
 +  __NOTOC__ 
   
−  ==Cauchy $\Delta$integral==
 +  Let $\mathbb{T}$ be a [[time scale]]. Delta integration is defined as the inverse operation of [[delta derivativedelta differentiation]] in the sense that if $F^{\Delta}(t)=f(t)$, then 
−  Let $\mathbb{T}$ be a [[time_scale  time scale]]. We say that $f$ is regulated if its rightsided limits exist (i.e. are finite) at all rightdense points of $\mathbb{T}$ and its leftsided limits exist (i.e. are finite) at all leftdense points of $\mathbb{T}$. We say that $f$ is predifferentiable with region of differentiation $D$ if $D \subset \mathbb{T}^{\kappa}$, $\mathbb{T}^{\kappa} \setminus D$ is countable with no rightscattered elements of $\mathbb{T}$, and $f$ is $\Delta$differentiable at each $t \in D$.  +  $$\displaystyle\int_s^t f(\tau) \Delta \tau = F(t)F(s).$$ 
−  Now suppose that $f$ is regulated. It is known that there exists a function $F$ which is predifferentiable with region of differentiation $D$ such that $F^{\Delta}(t)=f(t)$. We define the indefinite integral of a regulated function $f$ by
 
−  $$\displaystyle\int f(t) \Delta t = F(t)+C$$
 
−  for an arbitrary constant $C$.
 
−   
−  Now we define the definite integral, i.e. the Cauchy integral, by the formula
 
−  $$\displaystyle\int_s^t f(\tau) \Delta \tau = F(t)F(s)$$  
−  for all $s,t \in \mathbb{T}$.
 
−   
−  A function $F \colon \mathbb{T}\rightarrow \mathbb{R}$ is called an antiderivative of $f \colon \mathbb{T}\rightarrow \mathbb{R}$ if $F^{\Delta}(t)=f(t)$ for all $t \in \mathbb{T}^{\kappa}$. It is known that all rdcontinuous functions possess an antiderivative, in particular if $t_0 \in \mathbb{T}$ then $F$ defined by
 
−  $$F(t) = \displaystyle\int_{t_0}^t f(\tau) \Delta \tau$$
 
−  is an antiderivative of $f$.
 
−   
−  ==Riemann $\Delta$integral==
 
−   
−  ==Lebesgue $\Delta$integral==
 
   
 ==Properties of $\Delta$integrals==   ==Properties of $\Delta$integrals== 
Latest revision as of 23:46, 22 August 2016
Let $\mathbb{T}$ be a time scale. Delta integration is defined as the inverse operation of delta differentiation in the sense that if $F^{\Delta}(t)=f(t)$, then
$$\displaystyle\int_s^t f(\tau) \Delta \tau = F(t)F(s).$$
Properties of $\Delta$integrals
Delta integral from t to sigma(t)
Delta integral is linear
Interchanging limits of delta integral
Delta integrals are additive over intervals
Integration by parts for delta integrals with sigma in integrand
Integration by parts for delta integrals with no sigma in integrand
Delta integral over degenerate interval
Modulus of delta integral
Delta integral of nonnegative function
Delta integral of delta derivative
Delta derivative of the delta integral
References