User:Temperal/The Problem Solver's Resource10
Introduction | Other Tips and Tricks | Methods of Proof | You are currently viewing page 10. |
Integrals
This section will cover integrals and related topics, the Fundamental Theorem of Calculus, and some other advanced calculus topics.
The there are two types of integrals:
Indefinite Integral
The indefinite integral, or antiderivative, is a partial inverse of the derivative. That is, if the derivative of a function is written as
, then the indefinite integral of
is
, where
is a real constant. This is because the derivative of a constant is
.
Notation
- The integral of a function
is written as
, where the
means that the function is being integrated in relation to
.
- Often, to save space, the integral of
is written as
, the integral of
as
, etc.
Rules of Indefinite Integrals
for a constant
and another constant
.
,
Definite Integral
The definite integral is also the area under a curve between two points and
. For example, the area under the curve
between
and
is
, as are below the x-axis is taken as negative area.
Definition and Notation
- The definite integral of a function between
and
is written as
.
, where
is the antiderivative of
. This is also notated $\int f(x)\,dx \eval^{b}_{a}$ (Error compiling LaTeX. Unknown error_msg), read as "The integral of
evaluated at
and
." Note that this means in definite integration, one need not add a constant, as the constants from the functions cancel out.
Rules of Definite Integrals
for any
.
Fundamental Theorem of Calculus
Let ,
,
. Suppose
is differentiable on the whole interval
(using limits from the right and left for the derivatives at
and
, respectively), and suppose that
is Riemann integrable on
. Then
.
In other words, "the total change (on the right) is the sum of all the little changes (on the left)." Back to page 9 | Continue to page 11