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Fundamental theorem of calculus

The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverses of each other. This means that if a 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.

Intuition Intuitively, the theorem simply says that if you know all the little instantanous changes in some quantity, then you may compute the overall change in the quantity by adding up all the little changes.

To get a feeling for the statement, we will start with an example. Suppose you travel in a straight line, starting at time t = 0, and with varying speeds. If d(t) denotes the distance from the origin at time t and v(t) denotes the speed at time t, then v(t) is the instantaneous rate of change of d and is therefore the derivative of d. Suppose you know only v(t) from your speedometer, and you want to recover d(t). The fundamental theorem of calculus says that you should integrate v in order to get d. And this is exactly what you would have done, even without knowing that theorem: record the speed at regular intervals, maybe after 1 minute, 2 minutes, 3 minutes and so on, and then multiply the first speed with 1 minute to get an estimate for the distance covered in the first minute, then multiply the second speed with 1 minute to get the distance covered in the second minute etc., and then add all the distances up. In order to get an even better estimate of your current distance, you need to record the speeds at shorter time intervals. The limit as the length of the intervals approaches zero is exactly the definition of the integral of v.

Formal statements Stated formally, the theorem says:

If the function f(x) is continuous on some interval[?] [a, b], then the function F(x) defined by:
<math>\int_a^x f(t) dt </math>
is differentiable on the whole interval and its derivative is f(x).

Here, t is a dummy variable[?] which is only used for the purposes of the integration operation.

Part II of the theorem gives an important method for computing the integral of the continuous function f:

If <math>F(x)</math> is any antiderivative of the function <math>f(x)</math> (i.e. if <math>F'(x) = f(x)</math>), then
<math>\int_a^b f(x) dx = F(b) - F(a)</math>

As an example, suppose you need to calculate

<math>\int_2^5 x^2 dx </math>
Here, <math>f(x) = x^2</math> and we can use <math>F(x) = 1/3 x^3</math> as antiderivate. Therefore:

<math>\int_2^5 x^2 dx = F(5) - F(2) = {125 \over 3} - {8 \over 3} = {117 \over 3} = 39</math>

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 <math>[a, b]</math> and <math>x_0</math> is a number in <math>[a, b]</math> such that <math>f</math> is continuous at <math>x_0</math>, then

<math>F(x) = \int_a^x f(t) dt</math>
is differentiable for <math>x = x_0</math> with <math>F'(x_0) = f(x_0)</math>.

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 : [a, b] -> U, the curve integral[?] can be computed as

<math>\oint_{\gamma} f(z) \,dz = F(\gamma(b)) - F(\gamma(a))</math>

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.

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