A

Area between two curves

Criticality: 3

The region enclosed by the graphs of two functions over a specified interval, calculated by integrating the absolute difference between the functions.

Example:

To find the area between two curves like $y=x^2$ and $y=\sqrt{x}$ , you would integrate $(\sqrt{x} - x^2)$ from their intersection points.

B

Bottom function

Criticality: 3

When calculating the area between two curves, this is the function whose graph has lesser y-values than the other function over the interval of integration.

Example:

For the area between $y=e^x$ and $y=x$ on $[0,2]$ , $y=x$ is the bottom function because its graph lies below $y=e^x$ throughout the interval.

D

Definite Integral for Area

Criticality: 3

A specific application of the definite integral used to compute the area of a region bounded by two or more curves, typically by integrating the difference between the upper and lower functions.

Example:

The formula $A = \int_a^b [f(x) - g(x)]dx$ is the definite integral for area, where $f(x)$ is the top function and $g(x)$ is the bottom function over the interval $[a,b]$ .

I

Intersection points

Criticality: 3

The x-values where the graphs of two functions meet, which are often used as the limits of integration when determining the area enclosed by the curves.

Example:

To find the area bounded by $y=\cos(x)$ and $y=\sin(x)$ on $[0, \pi/2]$ , you first find their intersection points by setting $\cos(x)=\sin(x)$ , which occurs at $x=\pi/4$ .

T

Top function

Criticality: 3

When calculating the area between two curves, this is the function whose graph has greater y-values than the other function over the interval of integration.

Example:

If you're finding the area between $f(x)=x+2$ and $g(x)=x^2$ on $[0,1]$ , $f(x)$ would be the top function because its graph is above $g(x)$ on that interval.

Glossary

Area between two curves

Bottom function

Definite Integral for Area

Intersection points

Top function