Chapter 5 – Integral Calculus: Area, Definite Integrals, FTC & Substitution

Core question: “How can we turn the geometric notion of area under a curve into an algebraic quantity we can calculate?”
Riemann rectangles
- Partition the interval $[a,b]$ into $n$ sub-intervals of equal (or variable) width $\Delta x_i$ .
- Choose a sample point $x_i^*$ in each sub-interval.
- Area of the $i$ -th rectangle: $f(x<em>i^*)\,\Delta x</em>i$ .
- Riemann sum: $\sum{i=1}^{n}f(xi^{*})\,\Delta x_{i}$
Left-endpoint, right-endpoint, and midpoint rules
- Left-endpoint sampling tends to underestimate if $f$ is increasing, overestimate if $f$ is decreasing.
- Right-endpoint sampling behaves oppositely.
- Midpoint usually provides a better approximation (error proportional to $1/n^2$ vs $1/n$ for the two endpoint rules).
Limit idea
- As $n\to\infty$ (i.e.
  $\max\Delta xi\to0$ ) the Riemann sums, if they converge, approach the definite integral:
- ${\displaystyle\lim{n\to\infty}\sum{i=1}^{n}f(xi^{*})\,\Delta xi=f(x)\,dx.}$
Over/under estimates visualized
- Example: $f(x)=x^2$ on $[0,1]$ .
- Left sum with $n=4$ rectangles: width $0.25$ , heights $0,0.0625,0.25,0.5625\Rightarrow\text{area}=0.21875$ (underestimate; true area $=1/3\approx0.3333$ ).
- Right sum: $0.0625+0.25+0.5625+1\Rightarrow\text{area}=0.46875$ (overestimate).
Philosophical note: Area (a geometric primitive) is reduced to limits of sums—one of the earliest demonstrations of “continuous quantity ⇒ limiting process.”

Definition
- A bounded function $f$ on $[a,b]$ is integrable (Riemann-integrable) if its Riemann sums converge to the same limit regardless of the choice of sample points.
Guarantee of integrability
- Every continuous function on a closed interval is integrable.
- Discontinuous functions can still be integrable if their discontinuities form a “small” set (e.g. a finite number of jump discontinuities or any set of measure 0, like the set of rationals).
- Example of an integrable discontinuous function: $f(x)=\begin{cases}1, & x\ne0,\\ x=0 & \end{cases}$ The integral equals $2.$
Definite vs. indefinite integrals
- Definite integral $\int_a^b f(x)\,dx$ ⇒ single number (net signed area).
- Indefinite integral $\int f(x)\,dx$ ⇒ family of antiderivatives $F(x)+C.$
Net area & orientation
- Area above the $x$ -axis contributes positively; area below contributes negatively.
- “Net area” ≠ “geometric area” unless $f\ge0$ .
Properties (useful for proofs & computations)
- Linearity: $\big(Af(x)+Bg(x)\big)dx=Af+B\int_{a}^{b}g.$
- Additivity on intervals: $f=f+\int_{c}^{b}$
- Reversal: $f=-f.$

Antiderivative
- A function $F$ satisfies $F'(x)=f(x).$ Not unique ⇒ “ $+C$ .”
Area function
- Fix a point $a$ ; define $F(x)=\int_a^x f(t)\,dt.$
- FTC Part I: $F'(x)=f(x)$ (differentiation undoes integration).
- Intuition: changing the upper limit from $x$ to $x+\Delta x$ adds a thin rectangle of height $f(x)$ and width $\Delta x$ .
Significance
- Gives a constructive way to build one antiderivative even when an explicit formula is unknown.
- Shows that “instantaneous rate of change of accumulated area” returns the original density function.
Role of $C$
- In a definite integral $\int_a^b f(x)dx=F(b)-F(a),$ the arbitrary constants cancel.

Computational statement
- If $F$ is any antiderivative of $f,$ then $\displaystyle\int_a^b f(x)\,dx = F(b)-F(a).$
Proof sketch
- Subdivide $[a,b]$ , apply Mean Value Theorem to $F,$ sum telescopes ⇒ limit.
Consequences
- Convert almost every integral-evaluation problem into an algebra problem of finding antiderivatives.
- Bridges differential calculus (finding $F$ ) with integral calculus (areas, total change).
Typical examples
- $\int0^{\pi}\sin x\,dx$ = $[-\cos x]0^{\pi}=(-\cos\pi)-(-\cos0)=2.$
- Average value of a function $f$ on $[a,b]$ : $f{}=\dfrac1{b-a}f(x)dx.$

Motivation
- Reverse of the Chain Rule: if $\dfrac{d}{dx}F(g(x)) = F'(g(x))g'(x),$ then $\int f(g(x))g'(x)dx$ can be simplified by letting $u=g(x).$
Indefinite integral procedure
1. Identify an inner function $u=g(x)$ whose derivative $g'(x)$ appears (up to a constant) in the integrand.
2. Write $du=g'(x)dx$ ⇒ replace $g'(x)dx\rightsquigarrow du$ .
3. Rewrite integral in terms of $u$ ; integrate; back-substitute $u=g(x).$
Definite integral version
- Change the bounds: if $x=a$ ⇒ $u=g(a),$ and $x=b$ ⇒ $u=g(b).$ Never mix $x$ -bounds with $u$ -integrand.
Examples
- Indefinite: $\int 3(1+3x)^2\,dx.$
- Let $u=1+3x \,\Rightarrow du=3dx.$
- Integral becomes $\int u^2\,du= \dfrac{u^3}{3}+C = \dfrac{(1+3x)^3}{3}+C.$
- Definite: $\int_0^1 2x\cos(x^2)\,dx.$
- $u=x^2,\;du=2x\,dx.$
- New bounds: $x=0\Rightarrow u=0,\;x=1\Rightarrow u=1.$
- Integral $=\int<em>0^1 \cos u\,du = \sin u\big|</em>0^1 = \sin1-0.$
Pitfalls & tips
- If $du$ differs by a constant factor, pull that constant outside the integral.
- When $u$ -substitution fails, consider algebraic rearrangement, trig identities, or alternative techniques (integration by parts, partial fractions, etc.).

Historical insight
- Newton & Leibniz discovered FTC in the late 1600s, formalizing the duality between accumulation and rate of change.
Real-world relevance
- Physics: displacement = integral of velocity; work = integral of force; electric charge = integral of current.
- Economics: consumer surplus = integral between demand curve and price; total cost from marginal cost.
Philosophical/ethical reflection
- The ability to compute “total impact” (area) from “instantaneous effect” (rate) underpins quantitative decision-making, from drug dosage to carbon-emission accounting.
Foundation for later topics
- Techniques of Integration, Numerical Integration (Trapezoid, Simpson), Improper Integrals.
- Differential equations: antiderivatives serve as solutions to $y'=f(x).$
- Multivariable calculus: Riemann sums generalize to double/triple integrals.