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Integral: properties

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Start with Integral: definition.

Recall, the Riemann integral $\int_{a}^{b} f(x) dx$ is the limit of the Riemann sums as $n \to \infty$. $$ \int_{a}^{b} f(x)\, dx = \lim_{n \to \infty}\sum_{i=1}^{n} f(c_{i})\Delta x. $$

Sums have very simple properties. Meanwhile, limits behave well with respect to addition. So, it's easy to derive properties of integrals.

Properties.

  1. Constant function rule:
    $$\int\limits_{a}^{b} c\, dx = c \left( b-a \right) $$
  2. Constant multiple rule:
    $$ \int_{a}^{b} c\, f(x)\, dx = c \, \int_{a}^{b} f(x)\, dx$$
  3. Sum Rule:
    $$\int_{a}^{b} \left( f(x) + g(x) \right) \, dx = \int_{a}^{b} f(x)\, dx + \int_{a}^{b} g(x)\, dx $$

These correspond to the rules of limits as well as the rules of differentiation. But there is no product rule or chain rule.

Integrals have several rules with no corresponding rules for derivatives though.

Additivity: $$\int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx = \int_{a}^{c} f(x) \, dx. $$

The area interpretation of additivity:

SumAreas.png

Or, $$\int_{a}^{b} f(x) \, dx (orange)+ \int_{b}^{c} f(x) \, dx (green)= \int_{a}^{c} f(x) \, dx (purple). $$ In other words, $$\text{Sum of the two areas } = \text{ The total area from } a \text{ to } c$$

The motion interpretation of additivity: $$\text{distance covered during the 1st hour } + \text{ distance covered during the 2nd hour } = \text{ distance during 2 hours}$$

Comparison Properties.

1. If $f(x) \geq 0$ on $[a,b]$ then $$ \int_{a}^{b} f(x) \, dx \geq 0 $$ (To prove, look at the RS, all terms $\geq 0$, $\Delta x \geq 0$, then use a comparison theorem for limits).

2. If $f(x) \geq g(x)$ on $[a,b]$ then $$\int_{a}^{b} f(x) \, dx \geq \int_{a}^{b} g(x) \, dx $$ Here: green LHS and orange RHS.

Comparison2.png

Motion interpretation: faster $\Rightarrow$ covers more distance.

What if we know only estimates of $f$? Like this: $$m \leq f(x) \leq M$$ for all $x$ in $[a,b]$. Then what?

Comparison3.png

We want to find an estimate for the orange area in terms of $a$, $b$, $m$, $M$: $$\int_{a}^{b} f(x) \, dx = \text{ area}.$$

Below, the yellow area on the left is less then the orange area. On the right, the green area is larger. But these areas are rectangles and are easy to compute.

AreaDifferences.png

Hence, we have.

3. If $$m \leq f(x) \leq M$$ for all $x$ in $[a,b]$, then $$m(b-a)\leq \int_{a}^{b} f(x) \, dx \leq N(b-a).$$

For the most important property see Derivative and integral: Fundamental Theorem of Calculus.