Formula & Calculator
Fundamental Theorem of Calculus (Definite Integral)
Connects differentiation and integration, allowing a definite integral to be evaluated directly from an antiderivative at the interval endpoints.
Interpretation
∫ₐᵇ f'(x) dx = F(b) − F(a). Evaluation of definite integral using antiderivative. Bridge between area and anti‑differentiation. Core calculus theorem.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| F | Antiderivative of f | |
| a, b | Lower and upper limits of integration |
What it means
Same as id=61. This theorem is the central result that makes calculus practical. It states that if F is an antiderivative of f, then the definite integral of f from a to b equals F(b) − F(a). This allows us to compute areas under curves, total accumulated change, and many other quantities without summing infinitely many rectangles. It is the key to solving ordinary differential equations and is the theoretical basis for numerical integration. The theorem is also used in probability (computing cumulative distribution functions) and in signal processing (Fourier transforms). Understanding this theorem is fundamental for any quantitative discipline.
Worked example
FTC (Definite Integral) – Two Examples
Real‑World| Parameter | Value |
|---|---|
| F(b) | F(3)=9 |
| F(a) | F(1)=1 |
| Parameter | Value |
|---|---|
| F(b) | sin(π/2)=1 |
| F(a) | sin(0)=0 |
Common mistakes
- Same as ID 61 – this is a duplicate entry.
- Note: Ensure you use the correct antiderivative and evaluate at bounds.
Applications
The Fundamental Theorem of Calculus (definite integral form) states that the definite integral of a derivative equals the difference of the original function at the endpoints. This is the workhorse of calculus, used to compute net change, area, and average values. Engineers apply it to calculate displacement from velocity, to design control systems (e.g., PID controllers), and to analyse signals in frequency domain. It is also used in calculating work, energy, and fluid flow. In economics, it helps compute cumulative surplus and total cost. This theorem is essential for all quantitative fields, providing a powerful link between differential and integral calculus, and enabling practical solutions to real‑world accumulation problems.
- Displacement and velocity analysis in kinematics
- Work, energy, and power calculations in mechanics
- Signal integration in control systems and signal processing
- Fluid flow rate and total flow volume computation
- Economic surplus and consumer/producer welfare calculations
Frequently Asked Questions
It states that if f is continuous on [a,b] and F is any antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) – F(a). This allows evaluation of definite integrals without limits of Riemann sums.
Reversing the order of subtraction: computing F(a) – F(b) instead of F(b) – F(a), which flips the sign of the result.
If the function has jump discontinuities, the integral may be split into intervals where the function is continuous. The theorem applies to each piece.
The definite integral gives the net signed area. The theorem provides a way to compute that area using antiderivatives.
It cancels out when subtracting F(b) – F(a), so we can ignore it.
F(x) = x³/3. Then F(1) – F(0) = 1/3 – 0 = 1/3.
If f is piecewise continuous, the theorem applies to each piece, and the total integral is the sum of the pieces.
Evaluate the proper integral and take the limit. For example, ∫₀^∞ e^(–x) dx = lim_{b→∞} (1 – e^(–b)) = 1.
- Computing areas, volumes, and work.
- Solving differential equations.
- Probability and statistics.
The function f must be integrable on [a,b] (which is guaranteed if f is continuous or piecewise continuous).