Glossary

Absolute Value (in area formula)

Criticality: 2

Used in the area formula $A=∫_c^d∣f(y)−g(y)∣dy$ to ensure that the difference between the functions is always positive, guaranteeing a positive area.

Example:

In $A = \int_{0}^{1} |y^2 - y| ,dy$ , the absolute value ensures we subtract the 'left' function from the 'right' function to get a positive width for each slice.

Area Between Curves

Criticality: 3

The total 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 $y=x$ , you would integrate the difference of the functions over their intersection interval.

Functions of y

Criticality: 2

Equations where the independent variable is y and the dependent variable is x, typically expressed in the form $x = f(y)$.

Example:

When calculating the area between $x = y^2$ and $x = y+2$ , you are working with functions of y.

Horizontal Slices

Criticality: 3

A method for approximating the area between curves by dividing the region into infinitesimally thin rectangles oriented horizontally, used when integrating with respect to y.

Example:

When finding the area between $x=f(y)$ and $x=g(y)$ , we use horizontal slices of width $dy$ and length $|f(y) - g(y)|$ .

Integrating with respect to y

Criticality: 3

The process of finding the antiderivative of a function where the variable of integration is y, denoted by $dy$ in the integral.

Example:

To find the area between $x=y^2$ and $x=y$ , you would set up an integral and perform integrating with respect to y.

Intersection Points

Criticality: 3

The coordinates where two or more curves meet, found by setting their equations equal to each other and solving for the variable.

Example:

Finding the intersection points of $y=x^2$ and $y=x$ involves solving $x^2=x$ , which gives $x=0$ and $x=1$ .

Interval of Integration

Criticality: 3

The specific range of values (on the x-axis for $dx$ or y-axis for $dy$) over which a definite integral is evaluated to find the area or other quantities.

Example:

If curves intersect at $y=0$ and $y=1$ , then $[0,1]$ is the interval of integration for finding the area between them.

Upper and Lower Limits (of integration)

Criticality: 3

The specific values at the top and bottom of the integral sign, representing the boundaries of the interval over which the function is integrated.

Example:

In $\int_{0}^{1} (y - y^2) ,dy$ , $1$ is the upper limit and $0$ is the lower limit, defining the range of y-values for the area calculation.