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
The fundamental theorem of calculus confirms this. If F(x) is the antiderivative for f(x), we see
identically vanishes.

Additivity. Likewise, if we have the integral be over [a,c] and a<b<c, then
Again, using the fundamental theorem of calculus we see
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
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].


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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