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 .
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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)."
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