Glossary

Accumulation of Change

Criticality: 2

The total amount of change in a quantity over an interval, determined by integrating its rate of change.

Example:

If a function describes the rate at which a population grows, the accumulation of change over a decade would tell you the total increase in population during that time.

Area Under the Curve

Criticality: 2

The region bounded by the graph of a function, the x-axis, and two vertical lines at the interval's endpoints, whose signed value is given by the definite integral.

Example:

If a graph shows the rate of water flowing into a pool, the area under the curve over a certain time period represents the total volume of water added.

Average Value of a Function

Criticality: 3

The average y-value of a continuous function over a given interval, calculated by integrating the function over that interval and dividing by the length of the interval.

Example:

To find the average value of a function representing the temperature of a metal rod over its length, you would integrate the temperature function along the rod and divide by the rod's length.

Continuous Function

Criticality: 2

A function whose graph can be drawn without lifting the pen, meaning there are no breaks, jumps, or holes in its domain.

Example:

The function $f(x) = e^x$ is a continuous function for all real numbers, allowing us to apply the Average Value Theorem.

Definite Integral

Criticality: 3

An integral with upper and lower limits, representing the net accumulated change or the signed area under the curve of a function over a specific interval.

Example:

Calculating the definite integral of a car's acceleration function from $t=0$ to $t=10$ seconds gives the total change in its velocity during that time.

Interval

Criticality: 2

A set of real numbers between two specified endpoints, often denoted as $[a,b]$ for a closed interval.

Example:

When analyzing the average velocity of a particle, we might consider its motion over the interval from $t=2$ seconds to $t=5$ seconds.

Limits of Integration

Criticality: 2

The upper and lower bounds (a and b) of a definite integral, which define the specific interval over which the function is integrated.

Example:

In the integral $\int_{0}^{2\pi} \sin(x) , dx$ , the values 0 and $2\pi$ are the limits of integration.

Glossary

Accumulation of Change

Criticality: 2

The total amount of change in a quantity over an interval, determined by integrating its rate of change.

Example:

If a function describes the rate at which a population grows, the accumulation of change over a decade would tell you the total increase in population during that time.

Area Under the Curve

Criticality: 2

The region bounded by the graph of a function, the x-axis, and two vertical lines at the interval's endpoints, whose signed value is given by the definite integral.

Example:

If a graph shows the rate of water flowing into a pool, the area under the curve over a certain time period represents the total volume of water added.

Average Value of a Function

Criticality: 3

The average y-value of a continuous function over a given interval, calculated by integrating the function over that interval and dividing by the length of the interval.

Example:

To find the average value of a function representing the temperature of a metal rod over its length, you would integrate the temperature function along the rod and divide by the rod's length.

Continuous Function

Criticality: 2

A function whose graph can be drawn without lifting the pen, meaning there are no breaks, jumps, or holes in its domain.

Example:

The function $f(x) = e^x$ is a continuous function for all real numbers, allowing us to apply the Average Value Theorem.

Definite Integral

Criticality: 3

An integral with upper and lower limits, representing the net accumulated change or the signed area under the curve of a function over a specific interval.

Example:

Calculating the definite integral of a car's acceleration function from $t=0$ to $t=10$ seconds gives the total change in its velocity during that time.

Interval

Criticality: 2

A set of real numbers between two specified endpoints, often denoted as $[a,b]$ for a closed interval.

Example:

When analyzing the average velocity of a particle, we might consider its motion over the interval from $t=2$ seconds to $t=5$ seconds.

Limits of Integration

Criticality: 2

The upper and lower bounds (a and b) of a definite integral, which define the specific interval over which the function is integrated.

Example:

In the integral $\int_{0}^{2\pi} \sin(x) , dx$ , the values 0 and $2\pi$ are the limits of integration.