From 11:00PM PDT on Friday, July 1 until 5:00AM PDT on Saturday, July 2, the Shmoop engineering elves will be making tweaks and improvements to the site. That means Shmoop will be unavailable for use during that time. Thanks for your patience!

# The Fundamental Theorem of Calculus Terms

## Get down with the lingo

### Fundamental Theorem Of Calculus

This theorem provides a relation between the derivative and the definite integral of a function.

### Second Fundamental Theorem Of Calculus

This theorem provides a method of differentiating an integral.

### Integration By Parts

A technique for performing integration where .

### Improper Integral

An definite integral  is called an improper integral when the limits of integration are infinite (a = -∞, b = ∞) or the function becomes unbounded in [a, b].

### Reimann Sum

The Reimann Sum of a function f(x) over an interval [a, b] is defined as  where xi-1 < xi< xi, and xi for i = 1,…,n divides the interval [a, b] into n subintervals.

### Left-Hand Sum

The Left-Hand Sum of a function f(x) over an interval [a, b] is defined as  where xi for i = 1,…,n divides the interval [a, b] into n subintervals.

### Right-Hand Sum

The Right-Hand Sum of a function f ( x ) over an interval [a,b] is defined as  where xi for i = 1,…,n divides the interval [a,b] into n subintervals.

### Midpoint Sum

The Midpoint Sum of a function f (x) over an interval [a, b] is defined as where , and xi for i = 1,…,n divides the interval [a, b] into n subintervals.

### Trapezoid Sum

The trapezoid sum is the average between the Left-Hand and Right-Hand Sum.

### Average Value Of A Function

Average value of a continuous function f on the closed interval [a, b] is defined as .

### Concavity

This describes whether the function is curving up, down or not curving at all.

### Critical Point

The derivative of the function at the critical point is 0.

### Inflection Point

The point (x-value) where the function changes concavity.

### Secant Line

A line joining two points on the graph of a function.

### Tangent Line

A line that touches the graph of a function f (x) at a point.

### Differentiability

If the limit exists, the function f (x) is differentiable at x = a.