Changes

The Fundamental Theorem of Calculus says that the two operations are inverses -- to find the area under the graph of f(x) between a and b, find the function g(x) whose derivative is f(x) (that is, find the ''antiderivative'' of f).  The area under the graph of f between x=a and x=b is just g(b)-g(a).

The Fundamental Theorem of Calculus says that the two operations are inverses -- to find the area under the graph of f(x) between a and b, find the function g(x) whose derivative is f(x) (that is, find the ''antiderivative'' of f).  The area under the graph of f between x=a and x=b is just g(b)-g(a).

As a mathematical statement the fundametal theorem of calculus read,

The antiderivative of a function is often called the ''indefinite integral''.  (Indefinite because the limits a and b haven't been specified.)  So, for example, the derivative of <math>\frac{x^3}{3}+7</math> is <math>x^2</math>.  From this it follows that the antiderivative of  

<math>x^2</math> could be <math>\frac{x^3}{3}+7</math>.  But note that the "7" in that formula was a red herring.  Adding any constant to a function doesn't change its derivative, so the antiderivative of <math>x^2</math> could have any constant added to it.  This arbitrary constant is usually written '''C''' and is called the "constant of integration".  The indefinite integral could be written:

:<math>\int^{b}_{a}\frac{d}{dx}F(x)dx=F(b)-F(a)</math>

:<math>\int x^2\ \mathrm{d}x = \frac{x^3}{3} + C\,</math>,

For many years calculus focused on finding the anti-derivative of a function in order to integrate it. However for such functions as <math>e^{x^{2}}</math> no such antiderivative exists.

The Fundamental Theorem of Calculus says that the area under the graph of <math>x^2</math> between a and b is the difference in the values of <math>\frac{x^3}{3}+C</math> between a and b. Note that the constant of integration cancels out.

 

 

:<math>\int_a^b x^2\ \mathrm{d}x = \frac{b^3}{3} - \frac{a^3}{3}\,</math>,

 

The above is a simplified "intuitive" treatment of calculus and of this theorem.  The actual "rigorous" proof, "rigorous" definitions of derivative and integral, and statement of the conditions under which the theorem is true, are beyond the scope of this article.

[[Category:Calculus]]

[[Category:Calculus]]

[[Category:Mathematics]]

[[Category:Mathematics]]