8.2. Integrals

Unfortunately primitive functions are not uniquely determined by the presented integrable function. Due to [8.1.2] (which is in fact an implication of the mean value theorem) however any two primitives of f on an interval differ only in constant addend, which will vanish if we compute the difference of two values of f. This is a crucial feature for the integrals to be introduced now.

For the subsequent text let I be an arbitrary interval.

Consider:

• Continuous functions on intervals are integrable ([8.1.5]). Hence the integral ${\displaystyle \underset{a}{\overset{b}{\int }}f}$  exists for each ${\displaystyle f\in {\mathcal{C}}^0(I)}$.

• The term "definite" integral indicates that for its calculation an appropriate primitive function has to be evaluated at the fixed boundaries a and b. With an indefinite integral we would omit this calculation so that the symbol ${\displaystyle \underset{}{\overset{}{\int }}f}$, or sometimes ${\displaystyle \underset{}{\int }f+c}$, denotes just some primitive of f. The fundamental theorem of calculus [8.2.13] however demonstrates that this notation conveys a content meaning as well.

• To save parentheses we stipulate that the operators ${\displaystyle \underset{a}{\overset{b}{\int }}}$ and ${\displaystyle {|}_a^b}$ should obtain a higher priority than the basic arithmetics. For example we write

  ${\displaystyle \underset{2}{\overset{5}{\int }}3{\mathrm{X}}^2+2\mathrm{X}={\mathrm{X}}^3+{\mathrm{X}}^2{|}_2^5}$  instead of  ${\displaystyle \underset{2}{\overset{5}{\int }}(3{\mathrm{X}}^2+2\mathrm{X})=({\mathrm{X}}^3+{\mathrm{X}}^2){|}_2^5}$
  
   

• The integral [8.2.1] is often denoted by ${\displaystyle \underset{a}{\overset{b}{\int }}f(x)dx}$ or ${\displaystyle \underset{a}{\overset{b}{\int }}fd\mathrm{X}}$. This notation originates in an alternative approach to integrals where area measurement is the starting point and has just a symbolic meaning. We will dwell on this only in part 8.4.

  Beyond that however ${\displaystyle f(x)dx}$ and ${\displaystyle fd\mathrm{X}}$ resp. are introduced as a new objects to calculus, so called differential forms

  i

  We consider only differential forms of degree 1 in this case which are introduced as follows: Let ${\displaystyle {\omega }_x:ℝ\to ℝ}$ be a linear function for each ${\displaystyle x\in I}$, that means ${\displaystyle {\omega }_x(\alpha r+\beta s)=\alpha {\omega }_x(r)+\beta {\omega }_x(s)}$  for all ${\displaystyle r,s,\alpha ,\beta \in ℝ}$ Linear functions are often called homomorphisms and are understood as elements of ${\displaystyle Hom(ℝ,ℝ)}$. The function ${\displaystyle \omega :I\to I\times Hom(ℝ,ℝ)}$ defined by ${\displaystyle \omega (x)=(x,{\omega }_x)}$ is now called a differential form of degree 1 on I (a 1-form on I for short). We naturally assign a 1-form ${\displaystyle dg:x\mapsto (x,d_{x}g)}$ to each differentiable function ${\displaystyle g\in {\mathcal{D}}^1(I)}$ by setting ${\displaystyle d_{x}g(r)≔g^{\prime }(x)\cdot r}$ for ${\displaystyle x\in I}$. The function ${\displaystyle d_{x}g}$, obviously a homomorphism, is called the differential of g at x whereas ${\displaystyle d}$ is the differential of g. As an example we have ${\displaystyle d_x\mathrm{X}=\mathrm{X}}$ because: ${\displaystyle d_x\mathrm{X}(r)={\mathrm{X}}^{\prime }(x)\cdot r=1\cdot r=r}$. With ${\displaystyle {\omega }_x}$ being linear all its multiples are homomorphisms as well, thus for any function ${\displaystyle f:I\to ℝ}$ a differential form ${\displaystyle f\cdot \omega }$ is given by setting  ${\displaystyle f\cdot \omega (x)≔(x,f(x)\cdot {\omega }_x)}$. If f is integrable for example then ${\displaystyle fd\mathrm{X}}$ is an integrable differential form such that ${\displaystyle fd\mathrm{X}(x)=(x,f(x)\cdot \mathrm{X})}$.

  . [8.2.1] would then define the integral of the differential form ${\displaystyle fd\mathrm{X}}$ and the notation ${\displaystyle \underset{a}{\overset{b}{\int }}fd\mathrm{X}}$  now has a content meaning.


 

We start investigating integrals by looking at the behaviour at their boundaries. One important result is: integrals are additive at their boundaries.

1. allows to interrupt an integration at an arbitrary point c (c not even needs to be in between a and b!). Occasionally there are advantages from that. We could for instance calculate the integral ${\displaystyle \underset{-1}{\overset{1}{\int }}|\mathrm{X}|}$ without knowing any primitive of ${\displaystyle |\mathrm{X}|}$.We just split up the integration at 0 and thus have:

The next rules (integrals are linear in their integrands) are just integral versions of some derivation rules, namely sum, difference and factor rule.

Consider:

• We may now integrate polynomials addendwise:

  ${\displaystyle \underset{a}{\overset{b}{\int }}\sum _{i=0}^n{a}_i{\mathrm{X}}^i=\sum _{i=0}^n{a}_i\underset{a}{\overset{b}{\int }}{\mathrm{X}}^i=\sum _{i=0}^n\frac{a_i}{i+1}{\mathrm{X}}^{i+1}{|}_a^b}$

  for example ${\displaystyle \underset{0}{\overset{1}{\int }}8{\mathrm{X}}^3+2{\mathrm{X}}^2-3=\frac{8}{4}{\mathrm{X}}^4{|}_0^1+\frac{2}{3}{\mathrm{X}}^3{|}_0^1-3\mathrm{X}{|}_0^1=2+\frac{2}{3}-3=-\frac{1}{3}}$.

Product and chain rule are transferable as well. The resulting intergration rules are dealt with in the next part.

We proved several non-trivial properties of differentiable function using the mean value theorem, a basic theorem of calculus and should expect similar strong results from its integral version.

Consider:

• The proof of [8.2.8] is actually the proof of the implication

  mean value theorem, differential version${\displaystyle \Rightarrow }$mean value theorem, integral version

  In fact however both versions are equivalent as the reverse

  mean value theorem, integral version${\displaystyle \Rightarrow }$mean value theorem, differential version

  is valid as well.

  Proof:  ? ${\displaystyle f\in {\mathcal{D}}^1([a,b])}$
   If ${\displaystyle f\in {\mathcal{D}}^1([a,b])}$ then ${\displaystyle f^{\prime }\in \mathcal{I}([a,b])}$. Due to [8.2.8] there is now an ${\displaystyle \tilde{x}\in ]a,b[}$ such that
   

  ${\displaystyle f(b)-f(a)=f\phantom{}{|}_a^b=\underset{a}{\overset{b}{\int }}{f}^{\prime }=(b-a)\cdot f^{\prime }(\tilde{x})}$
  
   

With its integral version [8.2.8] the mean value theorem controls how integrals are affected by properties of their integrands. One example describes integration as a monotone process.

As the above proof obviously remains valid if we replace ${\displaystyle \le }$ by ${\displaystyle <}$ we see that integration is even strictly monotone.

The monotony itself will yield an important estimate that is needed if we are going to interchange integral and absolute value.

Consider:

• In general the estimate [8.2.11] could not be tightened to =. For example, as shown above, ${\displaystyle \underset{-1}{\overset{1}{\int }}|\mathrm{X}|=1}$, but  ${\displaystyle |\underset{-1}{\overset{1}{\int }}\mathrm{X}|=0}$.

  If however the algebraic sign of f is homogeneous on ${\displaystyle [a,b]}$, that means ${\displaystyle f(x)\ge 0}$ or ${\displaystyle f(x)\le 0}$ resp. for all ${\displaystyle x\in [a,b]}$, [8.2.11] may be replaced by the identity

  ${\displaystyle |\underset{a}{\overset{b}{\int }}f|=\underset{a}{\overset{b}{\int }}|f|}$

  [8.2.12]

  Proof:  According to [8.2.10] we have:

  1. ►  ${\displaystyle f(x)\ge 0}$ for all ${\displaystyle x\in [a,b]\Rightarrow \underset{a}{\overset{b}{\int }}f\ge 0}$, thus: ${\displaystyle |\underset{a}{\overset{b}{\int }}f|=\underset{a}{\overset{b}{\int }}f=\underset{a}{\overset{b}{\int }}|f|}$

  2. ►  ${\displaystyle f(x)\le 0}$ for all ${\displaystyle x\in [a,b]\Rightarrow \underset{a}{\overset{b}{\int }}f\le 0}$, thus: ${\displaystyle |\underset{a}{\overset{b}{\int }}f|=-\underset{a}{\overset{b}{\int }}f=\underset{a}{\overset{b}{\int }}-f=\underset{a}{\overset{b}{\int }}|f|}$


 

The next statement is an important joining element between differential and integral calculus. Moreover with suitable techniques at hand it will allow us to calculate primitives, which puts the triviality "Calculating integrals needs a good grasp of finding primitives" upside down: "Finding primitives needs a good grasp of calculating integrals".

Consider:

• [8.2.13] is sometimes called the second fundamental theorem of calculus. The first one would have been our definition [8.2.1] if integrals had been introduced in the alternative way mentioned before.

• As a primitive the function g from [8.2.13] is of course differentiable. Occasionally this is put into the phrase The integral is a differentiable function of its upper limit.

• Sometimes we simply use the symbol ${\displaystyle \underset{c}{\overset{}{\int }}f}$  for the function g which makes the relationship to the the indefinite integrals, as indicated initially, more transparent. Also the notation ${\displaystyle (\underset{c}{\overset{}{\int }}f{)}^{\prime }=f}$  [8.2.13] (combined with ${\displaystyle \underset{c}{\overset{}{\int }}(f^{\prime })=f-f(c)}$ ) supports the phrasing "integration and differentiation (nearly) cancel each other out".

• ${\displaystyle g(c)=0}$ due to [8.2.3]. Thus for any integrable f  [8.2.13] produces just the one primitive of f that has a zero at c.

• With that property we see that not all primitives could be produced by the fundamental theorem. The primitive ${\displaystyle {\mathrm{X}}^2+1}$ of ${\displaystyle 2\mathrm{X}}$ for instance has no zero and thus would never come out of [8.2.13].

We proved in [8.1.15] that integrability tolerates uniform convergence. Now we will see that even the integral itself is compatible with that kind of convergence..

8.1.

8.3.