The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverses of each other. It is of such central importance in calculus that it is called the fundamental theorem for the entire field of study. This means that if a continuous function is first integrated and then differentiated, the original function is retrieved. An important consequence of this, sometimes called the second fundamental theorem of calculus, allows one to compute integrals by using an antiderivative of the function to be integrated. In his 2003 book (page 340), James Stewart credits the idea that led to the fundamental theorem to the English mathematician Isaac Barrow although the first known proof of the fundamental theorem was due to the Scottish mathematician James Gregory.

Intuition

---

Intuitively, the theorem simply states that the sum of infinitesimal changes in a quantity over time (or some other quantity) add up to the net change in the quantity.

To comprehend this statement, we will start with an example. Suppose a particle travels in a straight line with its position given by x(t) where t is time. The derivative of this function is equal to the infinitesimal change in quantity per infinitesimal change in time (of course, the derivative itself is dependent on time). Let us define this change in distance per time as the speed v of the particle. In Leibniz's notation:

$\backslash frac\{dx\}\{dt\}\; =\; v(t)$

Rearranging that equation, it is clear that:

$dx\; =\; v(t)\backslash ,dt$

By the logic above, a change in x, call it $\backslash Delta\; x$, is the sum of the infinitesimal changes dx. It is also equal to the sum of the infinitesimal products of the derivative and time. This infinite summation is integration; hence, the integration operation allows the recovery of the original function from its derivative. As one can reasonably infer, this operation works in reverse as we can differentiate the result of our integral to recover the original function.

Formal statements

---

Donald_Duck_Fundamental_Theorem_Calculus.jpg|thumb|310px|The theorem is perhaps the most well known Fundamental theorem in popular culture. To impress the judges in a contest Donald Duck figure skates the theorem on ice. The theorem is written differently and is slightly wrong: one of the equal signs should have been a minus sign; also, the number 1 seems to have been written instead of the prime symbol in superscript above the first f. Donald has left out the dx, which is usually frowned upon. Nothing in the notation indicates the vector analysis version either. The correct version for real functions should have been

$\backslash int\_b^af\text{'}(x)\backslash ,dx\; =\; f(a)-f(b)$

]] Stated formally, the theorem says the following.

Let f be a continuous real-valued function defined on a closed interval b. Let F be the function defined for x in b by

$F(x)\; =\; \backslash int\_a^x\; f(t)\backslash ,\; dt$

then

$F\text{'}(x)\; =\; f(x)\backslash ,$

for every x in b.

Let f be a continuous real-valued function defined on a closed interval b. Let F be a function such that

$f(x)\; =\; F\text{'}(x)\backslash ,$ for all x in b

then

$\backslash int\_a^b\; f(x)\backslash ,dx\; =\; F(b)\; -\; F(a)$.

Corollary

---

Let f be a real-valued function defined on a closed interval b. Let F be a function such that

$f(x)\; =\; F\text{'}(x)\backslash ,$ for all x in b

then

$F(x)\; =\; \backslash int\_a^x\; f(t)\backslash ,dt\; +\; F(a)$

and

$f(x)\; =\; \backslash frac\{d\}\{dx\}\; \backslash int\_a^x\; f(t)\backslash ,dt$.

Examples

---

As an example, suppose you need to calculate

$\backslash int\_2^5\; x^2\backslash ,\; dx$

Here, $f(x)\; =\; x^2$ and we can use $F(x)\; =\; (1/3)\; x^3$ as the antiderivative. Therefore:

$\backslash int\_2^5\; x^2\backslash ,\; dx\; =\; F(5)\; -\; F(2)\; =\; \{125\; \backslash over\; 3\}\; -\; \{8\; \backslash over\; 3\}\; =\; \{117\; \backslash over\; 3\}\; =\; 39.$

Or, more generally, suppose you need to calculate

$\{d\; \backslash over\; dx\}\; \backslash int\_0^x\; t^3\backslash ,\; dt$

Here, $f(t)\; =\; t^3$ and we can use $F(t)\; =\; \{1\; \backslash over\; 4\}\; t^4$ as the antiderivative. Therefore:

$\{d\; \backslash over\; dx\}\; \backslash int\_0^x\; t^3\backslash ,\; dt\; =\; \{d\; \backslash over\; dx\}\; F(x)\; -\; \{d\; \backslash over\; dx\}\; F(0)\; =\; \{d\; \backslash over\; dx\}\; \{1\; \backslash over\; 4\}\; x^4\; =\; x^3$

But this result would have been found much more easily as

$\{d\; \backslash over\; dx\}\; \backslash int\_0^x\; t^3\backslash ,\; dt\; =\; f(x)\; \{dx\; \backslash over\; dx\}\; -\; f(0)\; \{d0\; \backslash over\; dx\}\; =\; x^3.$

Proof

---

It is given that

$F(x)\; =\; \backslash int\_\{a\}^\{x\}\; f(t)\; dt$

Let there be two numbers x₁ and x₁ + Δx in b. So we have

$F(x\_1)\; =\; \backslash int\_\{a\}^\{x\_1\}\; f(t)\; dt$

and

$F(x\_1\; +\; \backslash Delta\; x)\; =\; \backslash int\_\{a\}^\{x\_1\; +\; \backslash Delta\; x\}\; f(t)\; dt$.

Subtracting the two equations gives

$F(x\_1\; +\; \backslash Delta\; x)\; -\; F(x\_1)\; =\; \backslash int\_\{a\}^\{x\_1\; +\; \backslash Delta\; x\}\; f(t)\; dt\; -\; \backslash int\_\{a\}^\{x\_1\}\; f(t)\; dt\; \backslash qquad\; (1)$.

It can be shown that

$\backslash int\_\{a\}^\{x\_1\}\; f(t)\; dt\; +\; \backslash int\_\{x\_1\}^\{x\_1\; +\; \backslash Delta\; x\}\; f(t)\; dt\; =\; \backslash int\_\{a\}^\{x\_1\; +\; \backslash Delta\; x\}\; f(t)\; dt$.

(The sum of the areas of two adjacent regions is equal to the area of both regions combined.)

Manipulating this equation gives

$\backslash int\_\{a\}^\{x\_1\; +\; \backslash Delta\; x\}\; f(t)\; dt\; -\; \backslash int\_\{a\}^\{x\_1\}\; f(t)\; dt\; =\; \backslash int\_\{x\_1\}^\{x\_1\; +\; \backslash Delta\; x\}\; f(t)\; dt$.

Substituting the above into (1) results in

$F(x\_1\; +\; \backslash Delta\; x)\; -\; F(x\_1)\; =\; \backslash int\_\{x\_1\}^\{x\_1\; +\; \backslash Delta\; x\}\; f(t)\; dt\; \backslash qquad\; (2)$.

According to the mean value theorem for integration, there exists a c in x₁ + Δx such that

$\backslash int\_\{x\_1\}^\{x\_1\; +\; \backslash Delta\; x\}\; f(t)\; dt\; =\; f(c)\; \backslash Delta\; x$.

Substituting the above into (2) we get

$F(x\_1\; +\; \backslash Delta\; x)\; -\; F(x\_1)\; =\; f(c)\; \backslash Delta\; x\; \backslash ,$.

Dividing both sides by Δx gives

$\backslash frac\{F(x\_1\; +\; \backslash Delta\; x)\; -\; F(x\_1)\}\{\backslash Delta\; x\}\; =\; f(c)$.

Notice that the expression on the left side of the equation is Newton's difference quotient for F at x₁.

Take the limit as Δx → 0 on both sides of the equation.

$\backslash lim\_\{\backslash Delta\; x\; \backslash to\; 0\}\; \backslash frac\{F(x\_1\; +\; \backslash Delta\; x)\; -\; F(x\_1)\}\{\backslash Delta\; x\}\; =\; \backslash lim\_\{\backslash Delta\; x\; \backslash to\; 0\}\; f(c)$

The expression on the left side of the equation is the definition of the derivative of F at x₁.

$F\text{'}(x\_1)\; =\; \backslash lim\_\{\backslash Delta\; x\; \backslash to\; 0\}\; f(c)\; \backslash qquad\; (3)$.

To find the other limit, we will use the squeeze theorem. The number c is in the interval x₁ + Δx, so x₁ ≤ c ≤ x₁ + Δx.

Also, $\backslash lim\_\{\backslash Delta\; x\; \backslash to\; 0\}\; x\_1\; =\; x\_1$ and $\backslash lim\_\{\backslash Delta\; x\; \backslash to\; 0\}\; x\_1\; +\; \backslash Delta\; x\; =\; x\_1$.

Therefore, according to the squeeze theorem,

$\backslash lim\_\{\backslash Delta\; x\; \backslash to\; 0\}\; c\; =\; x\_1$.

Substituting into (3), we get

$F\text{'}(x\_1)\; =\; \backslash lim\_\{c\; \backslash to\; x\_1\}\; f(c)$.

The function f is continuous at c, so the limit can be taken inside the function. Therefore, we get

$F\text{'}(x\_1)\; =\; f(x\_1)\; \backslash ,$.

which completes the proof.

(Leithold et al, 1996)

Alternative proof

---

This is a limit proof by Riemann sums.

Let f be continuous on the interval b, and let F be an antiderivative of f. Begin with the quantity

$F(b)\; -\; F(a)\backslash ,$.

Let there be numbers

x₁, ..., x_n

such that

.

It follows that

$F(b)\; -\; F(a)\; =\; F(x\_n)\; -\; F(x\_0)\; \backslash ,$.

Now, we add each F(x_i) along with its additive inverse, so that the resulting quantity is equal:

& = & ^*\,+\,^*\,+\,^* \, \end{matrix}

The above quantity can be written as the following sum:

$F(b)\; -\; F(a)\; =\; \backslash sum\_\{i=1\}^n-\; F(x\_\{i-1\})\backslash qquad\; (1)$

Next we will employ the mean value theorem. Stated briefly,

Let F be continuous on the closed interval b and differentiable on the open interval (a, b). Then there exists some c in (a, b) such that

$F\text{'}(c)\; =\; \backslash frac\{F(b)\; -\; F(a)\}\{b\; -\; a\}.$

It follows that

$F\text{'}(c)(b\; -\; a)\; =\; F(b)\; -\; F(a).\; \backslash ,$

The function F is differentiable on the interval b; therefore, it is also differentiable and continuous on each interval x_i-1. Therefore, according to the mean value theorem (above),

$F(x\_i)\; -\; F(x\_\{i-1\})\; =\; F\text{'}(c\_i)(x\_i\; -\; x\_\{i-1\}).\; \backslash ,$

Substituting the above into (1), we get

$F(b)\; -\; F(a)\; =\; \backslash sum\_\{i=1\}^n-\; x\_\{i-1\}).$

The assumption implies $F\text{'}(c\_i)\; =\; f(c\_i).$ Also, $x\_i\; -\; x\_\{i-1\}$ can be expressed as $\backslash Delta\; x$ of partition $i$.

$F(b)\; -\; F(a)\; =\; \backslash sum\_\{i=1\}^nx\_i)\backslash qquad\; (2)$

Notice that we are describing the area of a rectangle, with the width times the height, and we are adding the areas together. Each rectangle, by virtue of the Mean Value Theorem, describes an approximation of the curve section it is drawn over. Also notice that $\backslash Delta\; x\_i$ does not need to be the same for any value of $i$, or in other words that the width of the rectangles can differ. What we have to do is approximate the curve with $n$ rectangles. Now, as the size of the partitions get smaller and n increases, resulting in more partitions to cover the space, we will get closer and closer to the actual area of the curve.

By taking the limit of the expression as the norm of the partitions approaches zero, we arrive at the Riemann integral. That is, we take the limit as the largest of the partitions approaches zero in size, so that all other partitions are smaller and the number of partitions approaches infinity.

So, we take the limit on both sides of (3). This gives us

$\backslash lim\_\{\backslash |\; \backslash Delta\; \backslash |\; \backslash to\; 0\}\; F(b)\; -\; F(a)\; =\; \backslash lim\_\{\backslash |\; \backslash Delta\; \backslash |\; \backslash to\; 0\}\; \backslash sum\_\{i=1\}^nx\_i)\backslash ,$

Neither F(b) nor F(a) is dependent on ||Δ||, so the limit on the left side remains F(b) - F(a).

$F(b)\; -\; F(a)\; =\; \backslash lim\_\{\backslash |\; \backslash Delta\; \backslash |\; \backslash to\; 0\}\; \backslash sum\_\{i=1\}^nx\_i)$

The expression on the right side of the equation defines an integral over f from a to b. Therefore, we obtain

$F(b)\; -\; F(a)\; =\; \backslash int\_\{a\}^\{b\}\; f(x)\backslash ,dx$

which completes the proof.

Generalizations

---

We don't need to assume continuity of f on the whole interval. Part I of the theorem then says: if f is any Lebesgue integrable function on $b$ and $x\_0$ is a number in $b$ such that $f$ is continuous at $x\_0$, then

$F(x)\; =\; \backslash int\_a^x\; f(t)\backslash ,\; dt$

is differentiable for $x\; =\; x\_0$ with $F\text{'}(x\_0)\; =\; f(x\_0)$. We can relax the conditions on f still further and suppose that it is merely locally integrable. In that case, we can conclude that the function F is differentiable almost everywhere and F'(x)=f(x) almost everywhere. This is sometimes known as Lebesgue's differentiation theorem.

Part II of the theorem is true for any Lebesgue integrable function f which has an antiderivative F (not all integrable functions do, though).

The version of Taylor's theorem which expresses the error term as an integral can be seen as a generalization of the Fundamental Theorem.

There is a version of the theorem for complex functions: suppose U is an open set in C and f: U -> C is a function which has a holomorphic antiderivative F on U. Then for every curve γ : b -> U, the curve integral can be computed as

$\backslash int\_\{\backslash gamma\}\; f(z)\; \backslash ,dz\; =\; F(\backslash gamma(b))\; -\; F(\backslash gamma(a)).$

The fundamental theorem can be generalized to curve and surface integrals in higher dimensions and on manifolds.

The most powerful statement in this direction is Stokes' theorem.

References

---

• Larson, Ron, Bruce H. Edwards, David E. Heyd. Calculus of a single variable. 7th ed. Boston: Houghton Mifflin Company, 2002.
• Leithold, L. (1996). The calculus 7 of a single variable. 6th ed. New York: HarperCollins College Publishers.
• Malet, A, Studies on James Gregorie (1638-1675) (PhD Thesis, Princeton, 1989).
• Rosa, Don (1994) From Duckburg to Lillehammer, Donald Duck 283, Disney.
• Stewart, J. (2003). Fundamental Theorem of Calculus. In Integrals. In Calculus: early transcendentals. Belmont, California: Thomson/Brooks/Cole.

• Turnbull, H W (ed.), The James Gregory Tercentenary Memorial Volume (London, 1939)

Calculus | Mathematical theorems

Teorema fonamental del càlcul | Infinitesimalregningens hovedsætning | Fundamentalsatz der Analysis | Teorema fundamental del cálculo integral | Fundamenta teoremo de kalkulo | قضیه اساسی حسابان | Théorème fondamental de l'analyse | 미적분학의 기본정리 | Teorema fondamentale del calcolo integrale | המשפט היסודי של החשבון הדיפרנציאלי והאינטגרלי | ანალიზის ფუნდამენტური თეორემა | 微分積分学の基本定理 | Podstawowe twierdzenie rachunku całkowego | Teorema fundamental do Cálculo | Основная теорема анализа | Analyysin peruslause | Analysens fundamentalsats | ทฤษฎีบทมูลฐานของแคลคูลัส | Hesabın temel teoremi | 微积分基本定理