If differential calculus studies how quantities change, integral calculus studies how they accumulate. The central object — the definite integral — is simultaneously an area, a limit, and an antiderivative evaluated at two points. This guide builds each idea from first principles, mirroring the companion Differentials reference.
The Fundamental Theorem of Calculus is the deepest result: it shows that these two subjects — differentiation and integration — are inverse operations, two faces of the same coin.
The Fundamental Theorem —
\[ \int_a^b f(x)\,dx = F(b) - F(a) \quad \text{where} \quad F'(x) = f(x) \]
① Riemann Sums
Left, right, midpoint rules
Summation notation
Definite integral as limit
Interactive visualiser
② Antiderivatives
Indefinite integral
Constant of integration C
Basic antiderivative table
③ Fundamental Theorem
FTC Part 1 & Part 2
Net change theorem
Properties of integrals
④ Techniques
Substitution (u-sub)
Integration by parts
Partial fractions
Trig identities & improper
⑤ Applications
Area between curves
Volumes of revolution
Arc length
Work, centre of mass
⑥ Jargon Glossary
40+ terms defined
Etymology notes
Cross-references
No terms match.
Section One
Riemann Sums & the Definite Integral
Definition — Definite Integral
The definite integral of \(f\) from \(a\) to \(b\) is the limit of Riemann sums as the number of subintervals \(n \to \infty\) and their widths \(\Delta x \to 0\):
\[ \int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*)\,\Delta x \]
Geometrically: signed area between \(f\) and the x-axis over \([a,b]\).
n = 4 |
Rule: Left |
Approx. ≈ 0 |
True = 0
4
a =0
b =3
The Three Main Riemann Sum Rules
Rule
Sample Point \(x_i^*\)
Notes
Left
Left endpoint \(x_{i-1}\)
Overestimates for increasing f
Right
Right endpoint \(x_i\)
Underestimates for increasing f
Midpoint
Midpoint \(\bar x_i = \tfrac{x_{i-1}+x_i}{2}\)
Usually most accurate of the three
Trapezoid
Average of left & right
\(\tfrac{\Delta x}{2}[f(x_{i-1})+f(x_i)]\); exact for linear f
⚠ Signed Area
When \(f(x) < 0\) on part of \([a,b]\), the integral counts that region as negative area.
To find total (unsigned) area, split the integral at the zeros and take absolute values:
\(\int_a^b |f(x)|\,dx\).
Real World — Distance from Velocity
If \(v(t)\) is a car's velocity, \(\int_0^T v(t)\,dt\) is the displacement (signed distance).
\(\int_0^T |v(t)|\,dt\) is the total distance travelled.
A GPS odometer sums up tiny distance increments — a digital Riemann sum.
Sigma Notation for Riemann Sums
With \(n\) equal subintervals of width \(\Delta x = (b-a)/n\) and \(x_i = a + i\Delta x\):
\[ L_n = \sum_{i=0}^{n-1} f(x_i)\,\Delta x \quad \text{(Left)} \qquad R_n = \sum_{i=1}^{n} f(x_i)\,\Delta x \quad \text{(Right)} \]
Useful identities: \(\sum_{i=1}^n i = \tfrac{n(n+1)}{2}\), \(\sum_{i=1}^n i^2 = \tfrac{n(n+1)(2n+1)}{6}\).
Section Two
Antiderivatives & Indefinite Integrals
Definition — Antiderivative
A function \(F\) is an antiderivative of \(f\) on an interval if \(F'(x) = f(x)\) for all \(x\) in that interval. The indefinite integral collects all antiderivatives:
\[ \int f(x)\,dx = F(x) + C \]
where \(C\) is an arbitrary constant (the constant of integration).
⚠ The Constant C is Not Optional
Omitting \(+ C\) is a fundamental error. Every differentiation rule "loses" constants, so every antiderivative family has infinitely many members differing by constants. Initial conditions or boundary values pin down \(C\).
Table of Basic Antiderivatives
Function \(f(x)\)
Antiderivative \(F(x) + C\)
Rule used
\(x^n,\; n \neq -1\)
\(\dfrac{x^{n+1}}{n+1} + C\)
Power rule (reversed)
\(\dfrac{1}{x}\)
\(\ln|x| + C\)
Log rule
\(e^x\)
\(e^x + C\)
Exp rule
\(a^x\)
\(\dfrac{a^x}{\ln a} + C\)
General exp
\(\sin x\)
\(-\cos x + C\)
Trig
\(\cos x\)
\(\sin x + C\)
Trig
\(\sec^2 x\)
\(\tan x + C\)
Trig
\(\dfrac{1}{\sqrt{1-x^2}}\)
\(\arcsin x + C\)
Inverse trig
\(\dfrac{1}{1+x^2}\)
\(\arctan x + C\)
Inverse trig
Finding Antiderivatives — Interactive
f(x): x² |
F(x): x³/3 + C |
C = 0
0
f(x)
F(x)+C
Real World — Initial Value Problems
A ball is thrown upward. Acceleration: \(a(t) = -9.8\) m/s². Find position.
\(\int -9.8\,dt = -9.8t + C_1\) (velocity). If \(v(0) = 20\): \(C_1 = 20\), so \(v(t) = -9.8t + 20\).
\(\int v\,dt = -4.9t^2 + 20t + C_2\) (position). If \(s(0) = 0\): \(C_2 = 0\).
Section Three
The Fundamental Theorem of Calculus
FTC — Part 1 (Differentiation of an Integral)
If \(f\) is continuous on \([a,b]\), then the accumulation function
\[ g(x) = \int_a^x f(t)\,dt \]
is differentiable on \((a,b)\) and \(g'(x) = f(x)\). Integration "undoes" differentiation.
FTC — Part 2 (Evaluation Theorem)
If \(F' = f\) and \(f\) is continuous on \([a,b]\), then:
\[ \int_a^b f(x)\,dx = F(b) - F(a) = \Big[F(x)\Big]_a^b \]
This collapses an infinite Riemann sum into two evaluations of \(F\).
∫02 f(x) dx = —
0.00
2.00
Properties of Definite Integrals
Property
Statement
Reversal
\(\displaystyle\int_b^a f\,dx = -\int_a^b f\,dx\)
Zero width
\(\displaystyle\int_a^a f\,dx = 0\)
Linearity
\(\displaystyle\int_a^b [cf + g]\,dx = c\int_a^b f + \int_a^b g\)
If \(f \ge g\) on \([a,b]\) then \(\int_a^b f \ge \int_a^b g\)
MVT for integrals
\(\exists\, c \in (a,b)\) s.t. \(f(c) = \dfrac{1}{b-a}\int_a^b f\,dx\)
Chain Rule Version of FTC Part 1
When the upper limit is a function \(u(x)\):
\[ \frac{d}{dx}\int_a^{u(x)} f(t)\,dt = f(u(x)) \cdot u'(x) \]
Example: \(\dfrac{d}{dx}\int_0^{x^2} \sin t\,dt = \sin(x^2) \cdot 2x\).
⚠ Net Change vs Total Change
\(\int_a^b f'(x)\,dx = f(b) - f(a)\) is the net change of \(f\). For displacement: if velocity changes sign, the car may travel far but return close to start. Net displacement ≠ total distance.
Historical Note
The FTC was independently established by Isaac Newton (~1666, "inverse method of fluxions") and Gottfried Leibniz (~1675, notation ∫ and dx). Their priority dispute dominated mathematics for decades. Both formulations survive: Newton's dot notation for time, Leibniz's ∫ for everything else.
Section Four
Integration Techniques
Core Methods
Direct Antiderivative
Recognise the integral as a known form directly from the antiderivative table. Always check this first.
u-Substitution
Let \(u = g(x)\), so \(du = g'(x)\,dx\):
\[\int f(g(x))\,g'(x)\,dx = \int f(u)\,du\]
The chain rule in reverse.
Integration by Parts
\[\int u\,dv = uv - \int v\,du\]
Choose \(u\) via LIATE: Log, Inverse trig, Algebraic, Trig, Exponential.
Partial Fractions
Decompose rational \(\frac{P(x)}{Q(x)}\) into simpler fractions when \(\deg P < \deg Q\).
Each factor of \(Q\) gets its own term.
Trig Identities
Use \(\sin^2 x = \frac{1-\cos 2x}{2}\), \(\cos^2 x = \frac{1+\cos 2x}{2}\), and Pythagorean identities to simplify trig powers.
Improper Integrals
Replace infinite limits with a variable and take a limit:
\[\int_a^\infty f\,dx = \lim_{t\to\infty}\int_a^t f\,dx\]
u-Substitution — Step by Step
Strategy
Look for a function \(g(x)\) whose derivative \(g'(x)\) also appears (up to a constant).
Set \(u = g(x)\), compute \(du = g'(x)\,dx\).
Rewrite the integral entirely in terms of \(u\).
Integrate in \(u\), then substitute back \(x\) variables.
For definite integrals: change the limits to \(u\)-values and never substitute back.
Example
\(\displaystyle\int 2x\cos(x^2)\,dx\) → Let \(u = x^2\), \(du = 2x\,dx\)
\[ = \int \cos(u)\,du = \sin(u) + C = \sin(x^2) + C \]
Integration by Parts — Step by Step
Strategy (LIATE rule)
Identify \(u\) (higher priority in LIATE) and \(dv\) (the rest including \(dx\)).
Compute \(du = u'\,dx\) and \(v = \int dv\).
Apply: \(\int u\,dv = uv - \int v\,du\).
If the new integral is simpler, evaluate it. If same type appears, use algebra.
Example
\(\displaystyle\int x e^x\,dx\) → \(u = x\), \(dv = e^x dx\) → \(du = dx\), \(v = e^x\)
\[ = xe^x - \int e^x\,dx = xe^x - e^x + C = e^x(x-1) + C \]
Partial Fractions — Example
Example
\(\displaystyle\int \frac{1}{x^2-1}\,dx\) → Factor: \(\dfrac{1}{(x-1)(x+1)} = \dfrac{A}{x-1} + \dfrac{B}{x+1}\)
Solve: \(A = 1/2\), \(B = -1/2\).
\[ = \frac{1}{2}\ln|x-1| - \frac{1}{2}\ln|x+1| + C = \frac{1}{2}\ln\left|\frac{x-1}{x+1}\right| + C \]
Trig Substitution — When to Use
Expression
Substitution
Identity used
\(\sqrt{a^2 - x^2}\)
\(x = a\sin\theta\)
\(1 - \sin^2\theta = \cos^2\theta\)
\(\sqrt{a^2 + x^2}\)
\(x = a\tan\theta\)
\(1 + \tan^2\theta = \sec^2\theta\)
\(\sqrt{x^2 - a^2}\)
\(x = a\sec\theta\)
\(\sec^2\theta - 1 = \tan^2\theta\)
⚠ Convergence of Improper Integrals
\(\displaystyle\int_1^\infty \frac{1}{x^p}\,dx\) converges if and only if \(p > 1\).
The "p-test" is the single most useful convergence criterion. The harmonic series \(\int_1^\infty \frac{1}{x}\,dx = \infty\) diverges (just barely).
Section Five
Applications of Integration
Area Between Curves
\[ A = \int_a^b \bigl[f(x) - g(x)\bigr]\,dx \quad \text{where } f(x) \ge g(x) \text{ on } [a,b] \]
Area between curves: —
a=0
b=2
Step-by-Step
Find where the curves intersect — these are your limits (or use given limits).
Determine which function is on top over the interval.
If the curves switch, split the integral at crossing points.
Volumes of Revolution
Disk/Washer Method (rotation about x-axis)
\[ V = \pi \int_a^b \bigl[f(x)^2 - g(x)^2\bigr]\,dx \]
Disk when \(g = 0\); washer when there's an inner radius \(g\).
Shell Method (rotation about y-axis)
\[ V = 2\pi \int_a^b x\,f(x)\,dx \]
Each thin vertical strip sweeps out a cylindrical shell.
Choosing a Method
Disks/washers: axis of rotation is parallel to the axis of integration. Slices perpendicular to axis.
Shells: axis of rotation is perpendicular to the axis of integration. Often avoids solving for the inverse.
Both methods give the same volume — choose whichever is simpler to integrate.
\(\bar x = \dfrac{\int x\,\rho(x)\,dx}{\int \rho(x)\,dx}\)
\(\rho\) = density function
Probability
\(P(a \le X \le b) = \displaystyle\int_a^b f(x)\,dx\)
\(f\) = probability density function
Real World — Consumer Surplus
In economics, if \(D(q)\) is the demand curve and \(p^*\) is the market price at quantity \(q^*\):
\[\text{Consumer Surplus} = \int_0^{q^*} [D(q) - p^*]\,dq\]
The area between the demand curve and the price line — the "extra value" consumers receive.
Section Six
Practice Problems
Problem 1 of 3 — Riemann
Challenge Problems
Stretch — Try Without Hints
Evaluate \(\displaystyle\int_0^1 x^2 e^x\,dx\) using integration by parts (twice).
Find the area enclosed by \(y = x^2\) and \(y = x + 2\).
Evaluate \(\displaystyle\int \frac{x^2+1}{x^3+3x}\,dx\) using partial fractions.
Determine whether \(\displaystyle\int_1^\infty \frac{\ln x}{x^2}\,dx\) converges; if so, find its value.
Find the volume of the solid formed by rotating \(y = \sqrt{x}\) about the x-axis over \([0,4]\).