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Fundamental Theorem of Calculus
Connects differentiation and integration: the definite integral of a derivative gives net change.
Interpretation
∫ₐᵇ f'(x) dx = f(b) − f(a). The definite integral of a derivative equals the net change of the original function. Connects differentiation and integration. Core of calculus.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| f(x) | Function | |
| a,b | Interval limits |
What it means
The Fundamental Theorem of Calculus is one of the most important results in mathematics, linking the two branches of calculus: differentiation and integration. It states that if f is continuous on [a,b] and F is its antiderivative (F'=f), then ∫ₐᵇ f(x) dx = F(b)−F(a). The theorem provides a practical way to evaluate definite integrals without resorting to limit sums. It also shows that integration is the inverse of differentiation. This theorem underpins many applications in physics, engineering, economics, and probability. For example, it allows computation of areas, volumes, work, and accumulated quantities. In engineering, it is used to calculate total displacement from velocity, energy from power, and charge from current. Understanding this theorem is essential for anyone studying calculus and its applications. It also forms the basis of the Navier‑Stokes equations, Maxwell’s equations, and many other models in science.
Worked example
Fundamental Theorem of Calculus – Two Examples
Real‑World| Parameter | Value |
|---|---|
| f'(x) | 2x |
| a | 0 |
| b | 1 |
| Parameter | Value |
|---|---|
| f'(t) | 3t² |
| a | 1 |
| b | 2 |
Common mistakes
- Continuity: f'(x) must be continuous on [a,b] for the theorem to apply.
- Antiderivative: The theorem requires finding an antiderivative F(x) such that F'(x)=f(x).
- Limits: Evaluate F(b) and F(a) correctly – the result is F(b) − F(a), not F(a) − F(b).
- Units: The result has the same units as the integrand times dx.
- Improper integrals: For unbounded intervals, use limits; the theorem applies to finite intervals.
Applications
The Fundamental Theorem of Calculus bridges differentiation and integration, stating that the definite integral of a derivative over an interval equals the change in the original function. This theorem is the cornerstone of calculus, enabling the evaluation of areas, volumes, and accumulated quantities using antiderivatives. Engineers and scientists use it to compute work done by a variable force, the total charge from a current, and the net change in a system's state. In physics, it underlies the derivation of equations of motion and energy conservation. In economics, it helps model cumulative costs and revenues. By applying this theorem, professionals can solve real‑world problems that require summing infinitesimal contributions, making it indispensable in fields from mechanics to finance. Its understanding is also essential for numerical integration methods used in computer simulations and engineering design.
- Calculation of areas, volumes, and centroid locations
- Work and energy computations in physics and engineering
- Accumulation of quantities (charge, mass, heat) over time or space
- Economic modelling of cumulative profit, cost, and revenue
- Foundation for numerical integration and computational simulations
Frequently Asked Questions
The Fundamental Theorem of Calculus connects differentiation and integration. It has two parts: Part 1 states that if f is continuous on [a,b], then F(x) = ∫ₐˣ f(t) dt is differentiable and F'(x) = f(x). Part 2 states that ∫ₐᵇ f(x) dx = F(b) – F(a), where F is any antiderivative of f.
- Part 1 (existence of antiderivatives) says that every continuous function has an antiderivative (given by the integral function).
- Part 2 (evaluation of definite integrals) provides a practical way to compute definite integrals using antiderivatives.
The function f must be continuous on the closed interval [a, b]. If there are discontinuities, the integral may still exist if they are removable or if the function is integrable, but the theorem as stated requires continuity.
The definite integral ∫ₐᵇ f(x) dx gives the net signed area between the curve y = f(x) and the x‑axis from x = a to x = b. The theorem allows us to compute this area by evaluating an antiderivative at the endpoints, without having to sum infinitely many rectangles.
When evaluating a definite integral using the theorem, the constant of integration cancels out: F(b) + C – (F(a) + C) = F(b) – F(a). Thus, we can ignore the constant for definite integrals.
If the function is piecewise continuous, split the integral at the breakpoints, apply the theorem to each piece, and sum the results. For example, ∫₀² f(x) dx = ∫₀¹ f₁(x) dx + ∫₁² f₂(x) dx.
- Physics: displacement from velocity (∫v dt) and work from force (∫F dx).
- Probability: cumulative distribution functions from probability density functions.
- Economics: total cost from marginal cost.
An indefinite integral (antiderivative) is a family of functions F(x) + C. A definite integral is a number representing the net area under a curve over a specific interval. The theorem links the two.
Improper integrals are evaluated as limits of proper integrals. The theorem applies to the proper integral, and then the limit is taken. For example, ∫ₐ^∞ f(x) dx = lim_{b→∞} ∫ₐᵇ f(x) dx = lim_{b→∞} [F(b) – F(a)].
- Forgetting that f must be continuous on the interval.
- Using an antiderivative that is not valid on the entire interval.
- Reversing the order of subtraction (F(a) – F(b) instead of F(b) – F(a)), which flips the sign.
- Applying the theorem to integrals that require substitution or integration by parts without performing those steps first.