The Definite Integral

The limit definition

Properties from the definition

Evaluating from the definition

The mean value theorem for integrals

The definite integral of f from a to b measures what geometric quantity?

Which equation expresses the additivity of the definite integral over adjacent intervals?

According to the convention for reversed limits, what is the integral from 3 to 1 of f(x) dx in terms of the integral from 1 to 3?

On [0, 1], the function f satisfies 0 at most f(x) at most 1 everywhere. What bound does the comparison property give for the integral from 0 to 1 of f?

Direct evaluation of the integral of x^2 from 0 to 1 uses the sum of the first n squares. What is that formula?

The average value of a continuous function f on [a, b] is defined as 1/(b - a) times the integral. What does the mean value theorem for integrals add to this definition?

The integral from a to b of a constant c equals c times the interval length. What is the integral from 1 to 4 of 5 dx? Answer with a number.

The integral from 2 to 2 of any integrable function f equals what number? Answer with a number.

On [0, 2], f(x) = x^2 and g(x) = x satisfy f at most g. Which is larger, the integral of f or the integral of g? Answer with f or g.

The triangle inequality in integral form says |integral of f| is at most the integral of |f|. As a three-letter abbreviation, what is this inequality commonly called? Answer in lowercase.