An integral is an operation on a function which describes the total signed area under it's graph.

Notation

Where is the "lower bound", is the "upper bound", is the function being integrated (called the integrand), and is the differential which describes which variable is being integrated with respect to.

Definition

The integral for the area over the interval can be described using a Riemann sum where the number of rectangles () approaches infinity, represented below using a limit.

Where is a partition of and , and is some -value arbitrarily picked in the interval .

Properties

Calculating

Graphically

Integrals can be estimated using a Riemann sum.

Symbolically

The fundamental theorem of calculus can be used to calculate the value of an integral.