Properties of the Integral

1. Introduction. There are about 4 properties we expect the integral to obey. We expect an integral over a zero width region should vanish
(1)\displaystyle\int^{a}_{a}f(x)\,\mathrm{d}x=0
The fundamental theorem of calculus confirms this. If F(x) is the antiderivative for f(x), we see
\displaystyle\int^{a}_{a}f(x)\,\mathrm{d}x=F(a)-F(a)=0
identically vanishes.

Additivity. Likewise, if we have the integral be over [a,c] and a<b<c, then
(2)\displaystyle\int^{b}_{a}f(x)\,\mathrm{d}x+\int^{c}_{b}f(x)\,\mathrm{d}x=\int^{c}_{a}f(x)\,\mathrm{d}x
Again, using the fundamental theorem of calculus we see
\displaystyle\int^{b}_{a}f(x)\,\mathrm{d}x+\int^{c}_{b}f(x)\,\mathrm{d}x=F(b)-F(a)+F(c)-F(b)=F(c)-F(a)
and
\displaystyle\int^{c}_{a}f(x)=F(c)-F(a)
Setting equals to equals yields (2).

This should make sense, since the intuition behind integration is consider a “continuous sum” of thin rectangular strips between y=0 and y=f(x). The width of each strip is \mathrm{d}x (“infinitesimal” wink wink), and we just keep adding more and more strips. In a sense this is just the associativity of addition.

Linearity of Integrand. Let f(x), g(x) be continuous functions on [a,b]. Let F(x), G(x) be their antiderivatives. Then
(3)\displaystyle\int^{b}_{a}\bigl[c_{1}f(x)+c_{2}g(x)\bigr]\,\mathrm{d}x=c_{1}\int^{b}_{a}f(x)\,\mathrm{d}x+c_{2}\int^{b}_{a}g(x)\,\mathrm{d}x
where c_1,c_2 are constants.

How do we see this? Again, by the fundamental theorem of calculus we have
\displaystyle  \begin{array}{rl}\displaystyle\int^{b}_{a}\bigl[c_{1}f(x)+c_{2}g(x)\bigr]\,\mathrm{d}x  &=\displaystyle\bigl(c_{1}F(b)+c_{2}G(b)\bigr)-\bigl(c_{1}F(a)+c_{2}G(a)\bigr)\\  &=\displaystyle c_{1}\bigl(F(b)-F(a)\bigr)+c_{2}\bigl(G(b)-G(a)\bigr)\end{array}
We see this is precisely the right hand side of (3), after invoking the fundamental theorem of calculus.

Special cases of linearity: f(x)+g(x) taking c_{1}=c_{2}=1; f(x)-g(x) taking c_{1}=1 and c_{2}=-1. Also kf(x) where k is a constant, take c_{1}=k, c_{2}=0.

Domination Property. On an interval [a,b] suppose f(x)\geq g(x) for each x\in[a,b]. Then
(4)\displaystyle\int^{b}_{a}f(x)\,\mathrm{d}x\geq \int^{b}_{a}g(x)\,\mathrm{d}x
The “proof” here is more intuitive: f(x)-g(x)\geq0 for each x\in[a,b]. Thus its integral over [a,b] is positive.

Inequalities. Take f(x) continuous on [a,b]. Let \min(f)\leq f(x)\leq\max(f) for each x\in[a,b]. Then
(5)\displaystyle\min(f)\cdot(b-a)\leq\int^{b}_{a}f(x)\,\mathrm{d}x\leq \max(f)\cdot(b-a)
This can follow from the domination property if we take g(x)=\min(f) and h(x)=\max(f) and observe g(x)\leq f(x)\leq h(x) for each x\in[a,b].

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About Alex Nelson

I like math. I like programming. Most of all, I love puzzles.
This entry was posted in Calculus, Definite Integral, Integral. Bookmark the permalink.

2 Responses to Properties of the Integral

  1. Pingback: [Index] Calculus of a Single Variable | My Math Blog

  2. Pingback: [Index] Calculus in a Single Variable | My Math Blog

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