Properties of Integrals

Suppose $f$ and $g$ are integrable on $[a,b]$.

Flipping the range

$$\int_a^bf=-\int_b^af$$

Addition

$f+g$ will be integrable on $[a,b]$.

$$\int_a^b(f\pm g)= \int_a^bf \pm \int_a^bg$$

Converse is not true.

Proof Hint

• Prove $f+g$ is integrable using:
  • $\sup(f+g)\le \sup(f) + \sup(g)$
  • $\inf(f+g)\ge \inf(f) + \inf(g)$
• Start with $U(f+g)$ and show $U(f+g)\le U(f)+U(g)$
• Start with $L(f+g)$ and show $L(f+g)\ge L(f)+L(g)$

Constant multiplication

Suppose $k\in\mathbb{R}$. $kf$ will be integrable $[a,b]$.

$$\int_a^bkf=k\int_a^bf$$

Converse is not true.

Proof Hint

• Prove for $k\ge 0$. Use $U-L\lt \frac{\epsilon}{k}$
• Prove for $k=-1$
• Using the above results, proof for $k<0$ is apparent

Bounds

If $m\le f(x) \le M$ on $[a,b]$:

$$m(b-a) \le \int_a^bf \le M(b-a)$$

If $f(x)\le g(x)$ on $[a,b]$:

$$\int_a^bf \le \int_a^bg$$

Modulus

$|f|$ will be integrable on $[a,b]$.

$$\Bigg|\int_a^bf\Bigg| \le \int_a^b|f|$$

Proof Hint

Start with $-|f|\le f \le |f|$. And integrate both sides. $$

Multiple

$fg$ will be integrable on $[a,b]$. Converse is not true.

Proof Hint

• Suppose $f$ is bounded by $k$
• Prove $f^2$ is integrable (Use $\frac{\epsilon}{2k}$)
• $fg$ is integrable because:

$fg=\frac{1}{2}\big[(f+g)^2 - f^2 - g^2\big]$

Max, Min

$\max(f,g)$ and $\min(f,g)$ are integrable.

Where $\max$ and $\min$ functions are defined as:

$\max(f,g)=\frac{1}{2}(|f-g|+f+g)$

$\min(f,g)=\frac{1}{2}(-|f-g|+f+g)$

Additivity

$\iff f$ is Riemann integrable on $[a,c]\text{ and } [c,b]$ where $c \in (a,b)$.

Proof Hint

• $\implies$: Use Cauchy criterion after defining these:
  • $P'=\set{c} \cap P$
  • $Q=P'\cap [a,c]$
  • $R=P'\cap [c,b]$
• $\impliedby$: Use cauchy criterion on $[a,c], [c,b]$ separately and then combine using a union partition

After the integrability is proven,

$$\int_a^b f = \int_a^c f + \int_c^b f$$

Proof Hint

1. Let $Q$ be a partition on $[a,c]$ and $R$ be a partition on $[c,b]$. And $P=Q \cap R$.

2. Prove the below using Cauchy criteria:

$$\int_a^b f \lt L(f;P) + \epsilon \;\; \implies \;\; \int_a^b f \le \int_a^c f + \int_c^b f$$

3. Prove the below using Cauchy criteria (by considering RHS):

$$\int_a^c f + \int_c^b f \le \int_a^b f$$