Math Glossary

Math Glossary

Anti-derivative, Definite Integral,

Indefinite Integral, Integral

Definition #1

The indefinite integral or anti-derivative of f(x) is any function whose derivative is f(x). The indefinite integral or anti-derivative of f(x) is denoted by

Note: Since you wouldn't know what constant term the anti-derivative would have, indefinite integrals always end with "+C" where C is an arbitrary constant unless more information is known to be able to find C.

Definition #2

If f(x)>0 on the interval [a,b], the area bounded by y=f(x), y=0, x=a and x=b is the definite integral of f(x) from a to b.

The definite integral of f(x) is denoted by

Definition #3

Using a continuous function f(x), if the interval is divided into n subintervals of equal length with endpoints x₀,x₁,x₂,...,x_n. Let c_i be any number in the i^th subinterval. That is, c₁ is in (x₀,x₁), c₂ is in (x₁,x₂), and so on. Then the definite integral of f(x) on (a,b) is

Note: The limit has to return the same number no matter how the c_i's are chosen for the limit to exist. If the c_i's are the left endpoints of the intervals, the sum is referred to a left sum. If the c_i's are the right endpoints of the intervals, the sum is referred to a right sum. If the c_i's are the midpoints of the intervals, the sum is referred to a midpoint sum.

Definition #4 (Reimann Integration)

If the interval is divided into n subintervals with endpoints x₀,x₁,x₂,...,x_n. This is called a partitions of the interval, ∆. Define ∆x_i as the length of the i^th subinterval with ||∆|| being the largest subinterval length. Also, let c_i be any number in the i^th subinterval. As in, c₁ is in (x₀,x₁), c₂ is in (x₁,x₂), and so on. Then the definite integral of f(x) on (a,b) is