Learn · With · Examples
A car’s speedometer and odometer are secretly the two halves of the most important theorem in calculus. One measures a rate, the other accumulates it — and the Fundamental Theorem of Calculus is simply the precise statement of how differentiation and integration undo each other. This guide builds it from that one everyday example, with interactive demos, worked problems, and code.
Contents
- The big idea — two operations, one relationship
- Formal statement — FTC Part 1 and Part 2
- Interactive: FTC Part 1 — differentiating accumulation
- Interactive: FTC Part 2 — evaluating definite integrals
- The odometer and speedometer model
- Interactive: watching area accumulate
- Why the two parts are really one theorem
- Worked example — projectile motion
- Common mistakes to avoid
- Real-world applications
- Differentiation vs integration — comparison table
- Code examples — Python
- Knowledge quiz
Section 01
The Big Idea — Two Operations, One Relationship
For most of calculus, derivatives and integrals feel like two completely separate tools. A derivative measures how fast something is changing at an instant — a slope. An integral measures the total accumulation of something over an interval — an area. They seem to answer opposite kinds of questions.
The Fundamental Theorem of Calculus (FTC) is the discovery that they are not separate at all. They are inverse operations — the same relationship that addition has with subtraction, or multiplication has with division. Differentiate an integral and you get back the original function. Integrate a derivative and you get back the original function (up to a constant). This single fact is why calculus works as one unified subject instead of two disconnected topics.
💡 The Key Idea
Every accumulation (integral) has an accumulation rate (derivative), and that rate is just the original function you were accumulating. Differentiation undoes integration. That’s the entire theorem — everything else is precise bookkeeping around that one idea.
Section 02
Formal Statement — FTC Part 1 and Part 2
Part 1 tells you that integration and differentiation are inverse processes. Part 2 is the practical payoff: it hands you a way to compute the exact value of a definite integral without summing infinitely many rectangles — just find an antiderivative and plug in the two endpoints.
⚠️ Continuity Requirement
Both parts require f to be continuous on the relevant interval. If f has a jump, a hole, or shoots to infinity somewhere in [a,b], the theorem doesn’t directly apply without extra care — this is why calculus courses spend so much time on continuity before ever reaching FTC.
Section 03
Interactive: FTC Part 1 — Differentiating Accumulation
Pick a function f(t). Watch its accumulation function F(x) get built via integration, then differentiated right back to f(x) — confirming Part 1 in each case.
Step 2 — Differentiate F(x): F'(x) = d/dx [x²/2] = x
Step 2 — Differentiate F(x): F'(x) = d/dx [x²+3x] = 2x + 3
Step 2 — Differentiate F(x): F'(x) = d/dx [sin(x)] = cos(x)
Section 04
Interactive: FTC Part 2 — Evaluating Definite Integrals
Four definite integrals, each solved the FTC way: find an antiderivative, then subtract F(a) from F(b). No rectangles, no limits of Riemann sums.
F(3) − F(0) = 27/3 − 0 = 9 − 0
F(4) − F(1) = (16+4) − (1+1) = 20 − 2
F(π) − F(0) = (−cos π) − (−cos 0) = 1 − (−1)
F(2) − F(0) = e² − 1 ≈ 7.389 − 1
Section 05
The Odometer and Speedometer Model
Every car dashboard already contains a working demonstration of FTC. The speedometer shows your instantaneous speed — a rate. The odometer shows your total distance traveled — an accumulation of that rate over time.
The odometer IS an accumulation function
odometer(t) = ∫₀ᵗ speed(τ) dτ + odometer(0). It’s literally integrating your speed over time to build up total distance.
The speedometer IS the derivative of the odometer
speed(t) = d/dt [odometer(t)]. At any instant, your speed is exactly how fast the odometer reading is changing. This is FTC Part 1, playing out on your dashboard.
Total distance = the odometer difference
Distance traveled between two times a and b = odometer(b) − odometer(a) = ∫ₐᵇ speed(t) dt. This is FTC Part 2 — you don’t need to track every instant of speed, just two odometer readings.
🚗 Why This Analogy Is Exact, Not Approximate
This isn’t a loose metaphor — it’s mathematically precise. Speed genuinely is the derivative of position, and position genuinely is the integral of speed. FTC isn’t describing something *like* a speedometer and odometer; a speedometer and odometer literally are a real-time physical instance of the theorem.
Section 06
Interactive: Watching Area Accumulate
Let f(t) = t (a straight line through the origin). As x grows, the shaded area under the line from 0 to x grows too — that shaded area IS F(x). Click through increasing values of x and watch the accumulation function build up, one snapshot at a time.
Notice the area doesn’t grow at a constant rate — it grows FASTER as x increases, because the height (f(x)=x) is also growing. That growing rate of area is exactly f(x) itself: FTC Part 1 in action.
Section 07
Why the Two Parts Are Really One Theorem
Part 1 and Part 2 can feel like two separate facts, but they’re two views of the exact same relationship, just pointed in opposite directions.
🔄 The Inverse-Operation Pattern
This mirrors √(x²) = x and (√x)² = x — squaring and square-rooting undo each other in both directions. Differentiation and integration have exactly this same inverse relationship, just operating on functions instead of numbers.
Section 08
Worked Example — Projectile Motion
A ball is thrown upward. Its velocity (accounting for gravity) is v(t) = −9.8t + 20 meters per second, where t is measured in seconds. Find the ball’s total displacement between t = 0 and t = 2 seconds.
Recognize this as FTC Part 2
Displacement is the integral of velocity. We need ∫₀² v(t) dt, and FTC Part 2 tells us to find an antiderivative and subtract endpoint values.
Find the antiderivative
F(t) = −4.9t² + 20t (check: F'(t) = −9.8t + 20 = v(t) ✓)
Evaluate F(2) and F(0)
F(2) = −4.9(4) + 20(2) = −19.6 + 40 = 20.4 F(0) = 0
Subtract
Displacement = F(2) − F(0) = 20.4 − 0 = 20.4 meters
F(t) = −4.9t² + 20t (antiderivative — position)
∫₀² v(t) dt = F(2) − F(0)
= (−4.9×4 + 40) − 0
= 20.4
The ball travels 20.4 meters (net) in the first 2 seconds.
⚠️ Displacement vs Total Distance
This calculation gives net displacement. If the ball goes up and then starts falling back down within those 2 seconds, the total distance traveled (odometer-style) would be larger, since up-then-down distances partially cancel in a plain integral. Total distance requires integrating |v(t)| instead.
Section 09
Common Mistakes to Avoid
Reversing F(a) and F(b)
The formula is F(b) − F(a), upper bound minus lower bound — never the other way around. Flipping it flips the sign of your answer.
Worrying about “+C”
For definite integrals, the constant of integration always cancels: (F(b)+C) − (F(a)+C) = F(b) − F(a). Any antiderivative works — you don’t need to find “the” one.
Ignoring discontinuities
If f has a vertical asymptote or jump inside [a,b], you can’t blindly apply FTC across it — the interval must be split or treated as an improper integral.
Confusing definite and indefinite
∫f(x)dx (indefinite) is a family of functions plus C. ∫ₐᵇf(x)dx (definite) is a single number. Only definite integrals get evaluated via FTC Part 2 directly.
Section 10
Real-World Applications
Physics — Motion
Displacement from velocity, velocity from acceleration, work done from a variable force — all direct FTC applications in mechanics.
Economics
Total cost is the integral of marginal cost. Consumer and producer surplus are computed as areas — definite integrals evaluated via FTC.
Medicine
Total drug absorbed into the bloodstream over time is the integral of the concentration-rate function — critical for dosage modeling.
Electrical Engineering
Total electric charge is the integral of current over time: Q = ∫ I(t) dt — a direct real-world use of FTC Part 2.
Probability
A cumulative distribution function (CDF) is the integral of a probability density function (PDF) — FTC Part 1 connects the two directly.
Biology
Total population growth over a period is the integral of the growth-rate function, used constantly in ecological and epidemiological modeling.
Section 11
Differentiation vs Integration — Comparison Table
| Aspect | Differentiation | Integration |
|---|---|---|
| Geometric meaning | Slope of the tangent line | Area under the curve |
| Physical meaning | Instantaneous rate of change | Total accumulation over an interval |
| Notation | f'(x) or dy/dx | ∫f(x)dx |
| Result type | Another function | A function (indefinite) or a number (definite) |
| Undoes | Integration | Differentiation |
| Everyday example | Speedometer reading | Odometer reading |
Section 12
Code Examples — Python
Verifying FTC symbolically with SymPy
from sympy import * x, t = symbols('x t') # FTC Part 1: differentiate an accumulation function f = t F = integrate(f, (t, 0, x)) # F(x) = x**2/2 check = diff(F, x) # should equal f print(F, "->", check) # x**2/2 -> x # FTC Part 2: evaluate a definite integral result = integrate(x**2, (x, 0, 3)) print(result) # 9 result2 = integrate(sin(x), (x, 0, pi)) print(result2) # 2
Numerical integration with SciPy (matching the analytic answer)
from scipy.integrate import quad import numpy as np # ∫₀² e^x dx — should be ~6.389 result, error = quad(lambda x: np.exp(x), 0, 2) print(f"{result:.4f}") # 6.3891 # Projectile displacement: v(t) = -9.8t + 20, from t=0 to t=2 displacement, error = quad(lambda t: -9.8*t + 20, 0, 2) print(f"{displacement:.1f} meters") # 20.4 meters
Section 13
Knowledge Quiz
Click a question to expand it, then pick your answer.
