Riemann Integration
Riemann Integration
The Riemann integral $\int_a^b f(x)\,dx$ is defined as the limit of Riemann sums. A bounded function is Riemann integrable iff its set of discontinuities has measure zero (Lebesgue's criterion). The Fundamental Theorem of Calculus (FTC) connects differentiation and integration: if $F'=f$, then $\int_a^b f=F(b)-F(a)$. This connection makes the integral computable and reveals that differentiation and integration are inverse operations.
Riemann Integrability
For $f:[a,b]\to\mathbb{R}$ bounded, a partition $P=\{a=x_0
Fundamental Theorem of Calculus
FTC Part 1: If $f:[a,b]\to\mathbb{R}$ is integrable and $F(x)=\int_a^x f(t)\,dt$, then $F$ is continuous on $[a,b]$ and $F'(x)=f(x)$ at every point where $f$ is continuous. FTC Part 2: If $f$ is continuous on $[a,b]$ and $G'=f$, then $\int_a^b f(x)\,dx=G(b)-G(a)$. Together they show differentiation and integration are inverse operations (up to constants) for continuous functions.
Example 1
Prove that every continuous $f:[a,b]\to\mathbb{R}$ is Riemann integrable.
Solution: By Cantor's theorem, $f$ is uniformly continuous on $[a,b]$. Given $\epsilon>0$, choose $\delta$ so $|x-y|<\delta\Rightarrow|f(x)-f(y)|<\epsilon/(b-a)$. Take partition $P$ with $\|P\|<\delta$. On each subinterval $[x_{i-1},x_i]$, $M_i-m_i<\epsilon/(b-a)$. Thus $U(f,P)-L(f,P)=\sum(M_i-m_i)\Delta x_i<\frac{\epsilon}{b-a}(b-a)=\epsilon$. So $f$ is integrable.
Example 2
Use FTC to evaluate $\int_0^1 x^3\,dx$ and $\frac{d}{dx}\int_0^{x^2}\sin t\,dt$.
Solution: $\int_0^1 x^3\,dx=[x^4/4]_0^1=1/4$. For $F(x)=\int_0^{x^2}\sin t\,dt$: let $u=x^2$, $G(u)=\int_0^u\sin t\,dt$, so $F(x)=G(x^2)$. By chain rule and FTC: $F'(x)=G'(x^2)\cdot 2x=\sin(x^2)\cdot 2x=2x\sin(x^2)$.
Practice
- Prove that monotone functions on $[a,b]$ are Riemann integrable using the criterion $U-L<\epsilon$.
- Evaluate $\int_0^\pi\sin x\,dx$ and $\int_0^1\ln x\,dx$ using the FTC (the second requires integration by parts).
- State Lebesgue's criterion for Riemann integrability: a bounded function is integrable iff its discontinuities form a set of measure zero.
- Prove the mean value theorem for integrals: if $f$ is continuous on $[a,b]$, then $\exists c\in(a,b)$ with $\int_a^b f=f(c)(b-a)$.
Show Answer Key
1. For monotone $f$ on $[a,b]$: choose partition with mesh $<\epsilon/(f(b)-f(a))$. Each subinterval contributes $\le(f(x_{i+1})-f(x_i))\cdot\text{mesh}$ to $U-L$. Summing telescopes: $U-L\le\text{mesh}\cdot|f(b)-f(a)|<\epsilon$.
2. $\int_0^\pi\sin x\,dx=[-\cos x]_0^\pi=2$. $\int_0^1\ln x\,dx=[x\ln x-x]_0^1=-1$ (using $\lim_{x\to0^+}x\ln x=0$).
3. A bounded function $f:[a,b]\to\mathbb{R}$ is Riemann integrable if and only if the set of discontinuities of $f$ has Lebesgue measure zero.
4. Let $F(x)=\int_a^x f$. By FTC, $F$ is differentiable with $F'=f$. By MVT, $\exists c$: $F(b)-F(a)=F'(c)(b-a)$, i.e., $\int_a^b f=f(c)(b-a)$.