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Fundamental Theorem of Calculus

Overview

Fundamental Theorem of Calculus :

For functions f(x),h(x),g(x) as in

h(x)=ddxf(x)

f(x)=∫xag(x)dx

→ First part states that h(x)=g(x); that is ddx∫xag(x)dx=g(x)*derivative of integral of a function is the function*

→ First part implies that to find f(x)=∫xag(x)dx, find the inverse relationship ddxf(x)=g(x) and thus, anti-derivative is defined.*integral of a function is anti-derivative of the function*

→ Second part states that ∫bah(x)dx=f(b)-f(a)

That is ∫baddxf(x)dx=f(b)-f(a).*definite integral of derivative of a function is the difference of function evaluated at the limits*

→ Since the second part is true for any value x in the interval [a,b], it implies that
∫xaddxf(x)dx=f(x)+c.*indefinite integral of derivative of a function is the function plus an initial value *

setting up

This topic is going to be little mathematically involved and firmly anchors some of the concepts in integrals. To prove Fundamental Theorem of Calculus, let us first understand "mean-value-theorem" and "squeeze theorem".

mean value theorem

Mean-Value-Theorem for Derivatives: For a continuous and differentiable function in the closed interval [a,b], there exists c such that a≤c≤b

ddxf(x)∣x=c =(f(b)-f(a))/(b-a)

The same can be written as

f′(c) =(f(b)-f(a))/(b-a)

The figure graphically explains the mean value theorem :

• (f(b)-f(a))/(b-a) is illustrated by the slope of hypotenuse of red triangle. In this, numeration and denominator are the two arms of right angle.

• ddxf(x)∣x=c is illustrated by the slope of tangent given by orange line. It is noted that. the tangent at c and the hypotenuse shown between a and b have same slope
there is at-least one point c in which slope of tangent equals slope of the hypotenuse of red triangle

mean value theorem again

Mean-Value-Theorem for integrals: For a continuous and integrable function in the closed interval [a,b], there exists c such that a≤c≤b

∫baf(x)dx=f(c)×(b-a)

The same can be written as

F(b)-F(a)=f(c)×(b-a)

where F(x) is the integral of f(x), given by F(x)=∫xaf(x)dx.
The figure graphically explains the mean value theorem:

• F(b)-F(a) is area under the curve shown in red.

• f(c)×(b-a) is the area of rectangle shown in green.

Area under the curve in red and the area of the rectangle in green are equal

There is at-least one point c in which area of the rectangle equals the area under the curve

Note: There is substantial overlap between red and green area in the figure, shown in dark greyish red color.

The name "mean-value-theorem" provide the clue as to what the theorem is about : one of the values in an interval is the mean for the interval.

squeeze theorem

Squeeze theorem: A value c is given such that a≤c≤b. If a-b tends to 0, then c=a=b.

The name of the theorem provide the clue as to what the theorem is about: When an interval a and b is squeezed to a=b, then a value between these two,c, is also squeezed to be equal to the two values.

First part

Given a continuous function f(x) in a closed interval [a,b] and differentiable in open interval (a,b).

h(x)=ddxf(x)

f(x)=∫xag(x)dx

First part of Fundamental Theorem of Calculus states that h(x)=g(x); that is ddx∫xag(x)dx=g(x)

Note: *Derivative of integral* of a function is the function.

The above means derivative is the inverse of integral.

First part also implies the abstraction "indefinite integral" or anti-derivatives.

∫g(x)dx=c+∫x0g(x)dx. *Note that the left hand side is indefinite integral and right hand side involves definite integral.*

In indefinite integral, f(x)=∫xag(x)dx, the function f(x) is derived from continuous aggregate or, equivalently, by limit of summation.

In anti-derivative, f(x)+c=∫g(x)dx, the function f(x) is derived from the inverse operation of derivative. That is, derive f(x) such that, derivative of f(x) is g(x).

In other words, f(x)+c=∫g(x)dx means the anti-derivative of g(x) is f(x) (even though the representation uses integral symbol). The name anti-derivatives is coined to specify this.

Second part

Given a continuous function f(x) in a closed interval [a,b] and differentiable in open interval (a,b).

h(x)=ddxf(x)

f(x)=∫xag(x)dx

Second part of Fundamental Theorem of Calculus states that ∫bah(x)dx=f(b)-f(a)

That is ∫baddxf(x)dx=f(b)-f(a).

*Integral of derivative* of a function is the function evaluated at the limits of integral.* (stated for definite integral.)*

This implies integral is the inverse of derivative.

Since the second part is true for any value x in the interval [a,b], it can be given as
∫xaddxf(x)dx=f(x)+c.

*Integral of derivative* of a function is the function with a constant of integration.* (stated for indefinite integral.)*

This also implies integral is the inverse of derivative.

proof 1

Proof for First part of Fundamental Theorem:
*Given a continuous function f(x) in a closed interval [a,b] and differentiable in open interval (a,b).
h(x)=ddxf(x)
f(x)=∫xag(x)dx
First part of Fundamental Theorem of Calculus states that h(x)=g(x); that is ddx∫xag(x)dx=g(x)*
h(x)

=ddxf(x)

=limδ→0f(x+δ)-f(x)δ

=limδ→0(∫x+δag(x)dx-∫xag(x)dx)/δ

=limδ→0(∫xag(x)dx+∫x+δxg(x)dx-∫xag(x)dx)/δ

=limδ→0(∫x+δxg(x)dx)/δ

=limδ→0(g(c)δ)/δ

=limδ→0g(c)

=g(x)

The above proves ddx∫xag(x)dx=g(x)

proof 2

Proof for Second part of Fundamental Theorem: *Given a continuous function f(x) in a closed interval [a,b] and differentiable in open interval (a,b).
h(x)=ddxf(x)
f(x)=∫xag(x)dx
Second part of Fundamental Theorem of Calculus states that ∫bah(x)dx=f(b)-f(a) That is ∫baddxf(x)dx=f(b)-f(a).*
To simplify the statements, take a=0

f(b)-f(0)

=n∑i=1(f(bin)-f(b(i-1)n)

=n∑i=1bnddxf(x)

=n∑i=1h(c)bn

=limn→∞n∑i=1h(c)bn

=limn→∞n∑i=1h(bin)bn

=∫b0h(x)dx

The above proves ∫xaddxf(x)dx=f(x)+c

in plain Language

The significance of Fundamental Theorem of Calculus:

The relationship between derivative and integral is understood as *inverse operations*.

Derivative and Integral are inverse operations.

• Derivative of integral of a function is the function.

• Integral of derivative of a function is the function or equivalent.

Summary of what we have learned:

• Differentiation, by the first principles, is the instantaneous-rate-of-change.

• Integration, by the first principles, is the continuous-aggregate.

• Integration is addition of change to an initial value.

• Indefinite integral is defined as a function of variable, where integration is carried out between 0 and variable x, along with an initial constant.

• Definite integral is defined as the quantity added for an interval a and b. This results in a numerical value.

• First part of fundamental theorem of calculus states that derivative of integral of a function is the function itself.

• Second part of fundamental theorem of calculus states that integral of derivative of a function is the function or an equivalent numerical result.

summary

**Fundamental Theorem of Calculus:** Given a continuous function f(x) in a closed interval [a,b] and differentiable in open interval (a,b).

h(x)=ddxf(x)

f(x)=∫xag(x)dx

First part states that h(x)=g(x); that is ddx∫xag(x)dx=g(x)

First part implies that to find f(x)=∫xag(x)dx, find the inverse relationship ddxf(x)=g(x) and thus, anti-derivative is defined.

Second part states that ∫bah(x)dx=f(b)-f(a)

That is ∫baddxf(x)dx=f(b)-f(a).

Since the second part is true for any value x in the interval [a,b], it implies that
∫xaddxf(x)dx=f(x)+c.

Outline

The outline of material to learn "Integral Calculus" is as follows.

• Detailed outline of Integral Calculus

→ __Application Scenario__

→ __Integration First Principles__

→ __Graphical Meaning of Integration__

→ __Definition of Integrals__

→ __Fundamental Theorem of Calculus__

→ __Algebra of Integrals__

→ __Antiderivatives: Standard results__

→ __Integration of Expressions__

→ __Integration by Substitution__

→ __Integration using Identities__

→ __Integration by Parts__

→ __Integration by Partial Fraction__

→ __Integration: Combination of Methods__