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The Fundamental Theorem of Calculus

April 6th, 2009 |

Author: Merrin

If we were forced to evaluate the Riemann sum every time we had to do a calculation, calculus wouldn’t have gotten too far off the ground. Just like our differentiation rules obscured the need to explicitly calculate limits there is a rule for integration. The fundamental theorem of calculus relates the area beneath a curve between two limits of integration to evaluations of the antiderivative at the limits. For a function, f(x), the antiderivative is another function written as F(x). Being an antiderivative means that F’(x)=f(x).

Theorem Fundamental Theorem of Calculus — First Part:

If f(x) is a continuous function on the closed interval x in [a,b] define the function F(x) by

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

Then F(x) is differentiable for x in (a,b) and F’(x)=f(x).

Proof First Part of the Fundamental Theorem of Calculus:

Take the functions F(x+Delta x) and F(x) and subtract them. Using the rules for adding areas between different intervals we find most of the area cancels out and we are left with an infinitesimal slice where a continuous function must be constant.

$F(x+\Delta x)-F(x)=\displaystyle \int _{a}^{x+\Delta x}f(t)dt -\displaystyle \int _{a}^{x}f(t)dt$

$\mathop{\lim }\limits_{\Delta x\to 0} F(x+\Delta x)-F(x)=\mathop{\lim }\limits_{\Delta x\to 0} \displaystyle \int _{x}^{x+\Delta x}f(t)dt$

$\mathop{\lim }\limits_{\Delta x\to 0} F(x+\Delta x)-F(x)=\mathop{\lim }\limits_{\Delta x\to 0} f(x^{*} )\Delta x$

Since the input of the function x^* is squeezed anywhere between x and

x+Delta x we can just write that function as f(x) since dx is infinitesimal.

$\mathop{\lim }\limits_{\Delta x\to 0} \cfrac{F(x+\Delta x)-F(x)}{\Delta x} =F'(x)=f(x)$

Theorem Fundamental Theorem of Calculus — Second Part:

If f(x) is a continuous function on the interval of x in [a,b], and F(x) is any antiderivative of f(x) then

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

Proof Second Part: We have from the first part of the Fundamental Theorem of Calculus that

$G(x)=\displaystyle \int _{a}^{x}f(x)dx$

Now define “any” antiderivative as F(x). The function F(x) can at most differ from G(x) by a constant because that is the most change allowed for taking a derivative to still get the integrand as a result.

$F(x)=G(x)+C$

If we evaluate the integral at x=a then we know G(a)=0 because the limits of integration are the same corresponding to the area of a line of width zero.

$F(a)=C$

${F(x)=G(x)+F(a)} \\ {G(x)=F(x)-F(a)}$

Now we evaluate at x=b

$G(b)=F(b)-F(a)$

Plugging this back into the original formula we have

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

The take home message is that in integration we are after the antiderivative. Given an antiderivative to the function f(x) the area between “a” and “b”, is calculated as F(b)-F(a). The fundamental theorem of calculus deals with what is called a definite integral. We are talking about the area under a curve between two limits. Sometimes it is useful to talk about just the antiderivative so we speak of indefinite integrals. Indefinite integrals equal any antiderivative plus an arbitrary constant. Since the derivative of a constant is zero, it is needed to write the most general antiderivative. Differentiating a constant plus an antiderivative is the same as just adding zero. The indefinite integral is written

$\displaystyle \int f(x)dx=F(x)+C$

Also a usual convention is that the capital letter of a lower case function

is the antiderivative, f(x) and F(x) for example. Definite integral calculations can be easily done having an indefinite integral. There is shorthand for evaluating the integrand at two different points. A vertical bar with two limits indicates

$\left. F(x)\right|_{a}^{b} =F(b)-F(a)$

After finding the antiderivative for a definite integral it is common to use this bar notation to finish off the calculation. It is generally considered harder to calculate the indefinite integral compared to the definite integral. This is partly because a definite integral is only a number while an indefinite integral is a function. If the limits of integration are on opposite sides of a singularity in the integrand, such as division by zero, then the definite integral may not exist, or further analysis is required.

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