Glossary

Absolute Value (in area calculation)

Criticality: 2

Used in the area formula $\int |f(x)-g(x)| dx$ to ensure that all regions between curves contribute positively to the total area, regardless of which function is momentarily 'on top'.

Example:

When $y=\sin(x)$ and $y=\cos(x)$ cross multiple times, using the absolute value in the integral ensures the total enclosed area is positive, not just the net signed area.

Area Between Curves

Criticality: 3

The measure of the region enclosed by two or more functions over a specified interval, calculated using definite integrals.

Example:

To find the area between curves $y=x^2$ and $y=\sqrt{x}$ , you would integrate the difference of the functions over their intersection interval.

Definite Integral

Criticality: 3

A calculus tool used to find the net signed area under a curve or between curves over a specific interval, represented by $\int_a^b f(x) dx$.

Example:

The definite integral $\int_0^2 (x^2-2x) dx$ calculates the net signed area between the curve $y=x^2-2x$ and the x-axis from $x=0$ to $x=2$ .

Functions of x

Criticality: 2

Equations where the dependent variable, typically 'y', is expressed explicitly in terms of the independent variable 'x', written as $y=f(x)$.

Example:

The equation $y = x^3 - 2x^2$ is a function of x, making it suitable for integration with respect to x.

Functions of y

Criticality: 2

Equations where the dependent variable, typically 'x', is expressed explicitly in terms of the independent variable 'y', written as $x=f(y)$.

Example:

The equation $x = y^2 + 1$ is a function of y, which is useful when integrating with respect to y for horizontal slices.

Horizontal Slices (Right-Left approach)

Criticality: 2

A method for calculating the area between curves where the region is divided into infinitesimally thin horizontal rectangles, integrating with respect to y.

Example:

To find the area enclosed by $x=y^2$ and $x=4$ , it's often easier to use horizontal slices and integrate $(4-y^2)dy$ from $y=-2$ to $y=2$ .

Intersection Points

Criticality: 3

The specific x-values (or y-values) where two or more functions meet, found by setting their equations equal to each other. These points typically define the limits of integration.

Example:

Finding the intersection points of $y=x^3$ and $y=x$ by solving $x^3=x$ helps determine the intervals for calculating the area between them.

Vertical Slices (Top-Bottom approach)

Criticality: 3

A method for calculating the area between curves where the region is divided into infinitesimally thin vertical rectangles, integrating with respect to x.

Example:

When finding the area between $y=x^2$ and $y=4$ , you would use vertical slices and integrate $(4-x^2)dx$ from $x=-2$ to $x=2$ .