Chapter: Integrals

These flashcards cover key vocabulary and concepts related to integrals, including definitions, methods of estimation, and important theorems.

Definite Integral

A definite integral gives the area under a curve between two points and results in a number.

Indefinite Integral

An indefinite integral represents a family of functions and includes a constant of integration.

Riemann Sum

The Riemann sum is an approximation of the integral by calculating the area of rectangles under the curve.

Upper Sum

An upper sum is an estimate of the area under a curve where rectangles are drawn above the curve.

Lower Sum

A lower sum is an estimate of the area under a curve where rectangles are drawn below the curve.

Midpoint Rule

In the midpoint rule, the heights of rectangles are determined by the function values at the midpoints of the bases.

Partition

A partition divides an interval into smaller subintervals for the purpose of calculating integrals.

Approximation Error

The approximation error measures the difference between the true value of an integral and its approximation.

Limit of a Riemann Sum

The limit of a Riemann sum as the number of rectangles approaches infinity gives the exact value of a definite integral.

Symmetric Integral Rule

For an even function, the integral over a symmetric interval is twice the integral from zero to the upper limit.

Mean Value Theorem for Definite Integrals

States that if a function is continuous on a closed interval, there exists at least one point at which the function's value is equal to the average value on that interval.

Average Value of a Function

The average value of a function over an interval is calculated using the integral of the function over the interval divided by the length of the interval.

Notation for Summation

The symbol Σ (sigma) represents the sum of a sequence of terms specified by a function.

Integration by Substitution

A method used to evaluate integrals by substituting a new variable to simplify the integral.