Kuzee
Integral Calculus (Single-Variable)

The Fundamental Theorem of Calculus

The fundamental theorem of calculus is the “soul” of the whole of calculus: it links two operations that seem entirely unrelated — differentiation and integration. Only once you understand this theorem do you really understand integration.

Historical Background: An “Accident” That Changed Mathematics

Before Newton and Leibniz, finding an area (integration) and finding a tangent (differentiation) were two independent kinds of problem:

  • The area problem: how do you find the area enclosed by a curve? The ancient Greeks used the method of exhaustion, and their successors used “sum up the pieces”.
  • The tangent problem: how do you find the slope of the tangent to a curve at a point? This is the origin of differential calculus.

In the seventeenth century Newton and Leibniz independently discovered a remarkable fact: the two problems are inverse operations to each other. This discovery turned countless area computations that had previously had to be attacked one at a time into a single uniform procedure. As later generations put it, “the fundamental theorem of calculus turned a craft into a science”.

The First Fundamental Theorem: Differentiating an Integral with Variable Limits

The upper-limit function

Let f(x)f(x) be continuous on [a,b][a, b]. The function

F(x)=∫axf(t) dtF(x) = \int_a^x f(t) \, dt

is called the upper-limit function. It represents the total area “accumulated from aa to xx”.

Theorem1

If f(x)f(x) is continuous on [a,b][a, b], then the upper-limit function F(x)F(x) is differentiable, and

F′(x)=f(x)F'(x) = f(x)

that is: differentiating the accumulated area gives the rate of change itself.

Idea of the proof (going back to the definition of the derivative):

F′(x)=lim⁡h→0F(x+h)−F(x)h=lim⁡h→0∫xx+hf(t) dthF'(x) = \lim_{h \to 0} \frac{F(x+h) - F(x)}{h} = \lim_{h \to 0} \frac{\int_x^{x+h} f(t)\, dt}{h}

By the mean value theorem for integrals, there is a ξ∈[x,x+h]\xi \in [x, x+h] such that ∫xx+hf(t) dt=f(ξ)⋅h\int_x^{x+h} f(t)\, dt = f(\xi) \cdot h, and hence

F′(x)=lim⁡h→0f(ξ)=f(x)F'(x) = \lim_{h \to 0} f(\xi) = f(x)

The Second Fundamental Theorem: The Newton-Leibniz Formula

Theorem2

Let f(x)f(x) be continuous on [a,b][a, b], and let G(x)G(x) be any antiderivative of f(x)f(x). Then

∫abf(x) dx=G(b)−G(a)\int_a^b f(x) \, dx = G(b) - G(a)

Idea of the proof: put F(x)=∫axf(t) dtF(x) = \int_a^x f(t)\, dt. By the first fundamental theorem, both F(x)F(x) and G(x)G(x) are antiderivatives of f(x)f(x), so G(x)−F(x)=CG(x) - F(x) = C (a constant). Setting x=ax = a gives C=G(a)C = G(a) (since F(a)=0F(a) = 0). Hence G(b)−G(a)=F(b)=∫abf(x) dxG(b) - G(a) = F(b) = \int_a^b f(x)\, dx, as required.

The Core Conclusion

∫abf(x) dx=[G(x)]ab=G(b)−G(a),G′(x)=f(x)\int_a^b f(x) \, dx = \left[ G(x) \right]_a^b = G(b) - G(a), \quad G'(x) = f(x)

Why It Is the “Fundamental” Theorem

The depth of the fundamental theorem of calculus lies in the fact that it unifies three threads:

  1. It unifies differentiation and integration: differentiation and integration are inverse operations.
  2. It unifies evaluation and area-finding: a definite integral no longer requires a complicated process of summing pieces — it is enough to find an antiderivative and substitute the limits.
  3. It unifies the local and the global: a local rate of change (the derivative), once integrated, restores the whole accumulated quantity.

It is precisely this theorem that raised “integration” from a geometric operation to a general-purpose method of calculation, and made it the cornerstone of every subject that uses calculus — physics, engineering, economics and beyond.

Typical Applications

Application 1: Differentiating an Integral with Variable Limits

Let F(x)=∫xx2sin⁡tt dtF(x) = \int_{x}^{x^2} \frac{\sin t}{t} \, dt. Find F′(x)F'(x).

an integral with variable limitsthe chain rulethe fundamental theorem of calculus

Approach:

Split the integral into the difference of two upper-limit functions, and then apply the chain rule.

Steps:

  1. F(x)=∫xx2sin⁡ttdt=∫0x2sin⁡ttdt−∫0xsin⁡ttdtF(x) = \int_{x}^{x^2} \frac{\sin t}{t} dt = \int_0^{x^2} \frac{\sin t}{t} dt - \int_0^{x} \frac{\sin t}{t} dt
  2. F′(x)=sin⁡(x2)x2⋅2x−sin⁡xxF'(x) = \frac{\sin(x^2)}{x^2} \cdot 2x - \frac{\sin x}{x}
  3. =2sin⁡(x2)x−sin⁡xx= \frac{2\sin(x^2)}{x} - \frac{\sin x}{x}

Answer: F′(x)=2sin⁡(x2)x−sin⁡xxF'(x) = \frac{2\sin(x^2)}{x} - \frac{\sin x}{x}

Application 2: Constructing an Antiderivative to Compute an Integral

Use the fundamental theorem of calculus to compute ∫01xex2dx\int_0^1 x e^{x^2} dx.

substitutionNewton-Leibniza definite integral

Approach:

First find an antiderivative by a substitution, and then substitute the limits.

Steps:

  1. Put u=x2u = x^2, so du=2xdxdu = 2x dx
  2. ∫xex2dx=12ex2+C\int x e^{x^2} dx = \frac{1}{2}e^{x^2} + C
  3. ∫01xex2dx=[12ex2]01=e−12\int_0^1 x e^{x^2} dx = [\frac{1}{2}e^{x^2}]_0^1 = \frac{e - 1}{2}

Answer: e−12\frac{e - 1}{2}

Summary

Symbols Used in This Article

SymbolTypePronunciation/ExplanationMeaning in This Article
∫axf(t) dt\int_a^x f(t)\, dtmathematical symbolintegral from a to xThe upper-limit function
F(x)F(x)mathematical symbolF of xThe upper-limit function
F′(x)F'(x)mathematical symbolF prime of xThe derivative of the upper-limit function
G(x)G(x)mathematical symbolG of xAn antiderivative of f(x)f(x)
[G(x)]ab[G(x)]_a^bmathematical symbolG of x from a to bThe notation for substituting the limits
ξ\xiGreek letterXiThe point whose existence the mean value theorem for integrals asserts
hhmathematical symbolhThe increment of the independent variable
lim⁡h→0\lim_{h \to 0}mathematical symbollimit as h goes to 0The limit as hh tends to zero
eemathematical symbolEuler’s numberThe base of the natural logarithm
sin⁡t\sin tmathematical symbolsine of tThe sine function
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