Glossary

Area between Two Polar Curves

Criticality: 3

The area of the region bounded by two polar curves, an outer curve $r_2$ and an inner curve $r_1$, calculated using the integral $\frac{1}{2}\int_a^b (r_2^2 - r_1^2) d\theta$.

Example:

Calculating the area between a circle $r=3$ and a cardioid $r=2+\cos(\theta)$ requires using the Area between Two Polar Curves formula.

Area of a Polar Region (single curve)

Criticality: 2

The area enclosed by a single polar curve from an angle $a$ to an angle $b$, calculated using the integral $\frac{1}{2}\int_a^b r^2 d\theta$.

Example:

To find the area of a single cardioid $r = 1 + \cos(\theta)$ , you would use the Area of a Polar Region (single curve) formula from $0$ to $2\pi$ .

Bounds (for polar area)

Criticality: 3

The specific angular values ($a$ and $b$) that define the start and end of the region for which the area is being calculated in polar coordinates. These are often found by setting the two polar equations equal to each other.

Example:

To find the area of a loop of a limacon, you'd need to determine the bounds (for polar area) where the curve intersects the origin or itself.

Double Angle Theorem

Criticality: 2

Trigonometric identities that express trigonometric functions of twice an angle in terms of functions of the original angle, such as $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$.

Example:

When integrating $\sin^2(\theta)$ , you might use the Double Angle Theorem to rewrite it as $\frac{1-\cos(2\theta)}{2}$ for easier integration.

Inside Radius ($r_1$)

Criticality: 3

In the context of finding the area between two polar curves, $r_1$ refers to the function representing the curve that is closer to the origin in the region of interest.

Example:

When calculating the area between $r=5$ and $r=3$ , $r=3$ would be your inside radius ( $r_1$ ).

Outside Radius ($r_2$)

Criticality: 3

In the context of finding the area between two polar curves, $r_2$ refers to the function representing the curve that is farther from the origin in the region of interest.

Example:

If you're finding the area between $r=5$ and $r=3$ , then $r=5$ would be your outside radius ( $r_2$ ).

Polar Coordinate System

Criticality: 2

A two-dimensional coordinate system where each point is determined by a distance from a reference point (the origin) and an angle from a reference direction (the positive x-axis).

Example:

Plotting a point like $(5, \pi/4)$ involves using the Polar Coordinate System.

Polar Curves

Criticality: 3

Curves defined by equations in polar coordinates, where points are specified by a distance from the origin (radius) and an angle from the positive x-axis.

Example:

The graph of $r = 2\cos( heta)$ is a polar curve representing a circle.

Polar Interval (a and b)

Criticality: 3

The angular range, represented by $a$ and $b$, over which the area of a polar region or between polar curves is calculated. These values are typically in radians.

Example:

For a full circle, the polar interval might be from $0$ to $2\pi$ .

Radian Values

Criticality: 3

A unit of angle measurement, where one radian is the angle subtended at the center of a circle by an arc equal in length to the radius. Calculus typically uses radians for angles.

Example:

When evaluating $\sin(\theta)$ in calculus, $\theta$ is almost always assumed to be in radian values.

Radius (r)

Criticality: 3

In polar coordinates, $r$ represents the distance of a point from the origin. In polar equations, it is typically a function of the angle $\theta$.

Example:

In the polar equation $r = 4\sin(\theta)$ , the radius $r$ changes with the angle $\theta$ .

Second Quadrant

Criticality: 2

The region in the Cartesian coordinate system where x-coordinates are negative and y-coordinates are positive, corresponding to angles between $\pi/2$ and $\pi$ in polar coordinates.

Example:

A point with polar coordinates $(r, \theta)$ where $\theta = 2\pi/3$ would lie in the Second Quadrant.

Glossary

Area between Two Polar Curves

Criticality: 3

The area of the region bounded by two polar curves, an outer curve $r_2$ and an inner curve $r_1$, calculated using the integral $\frac{1}{2}\int_a^b (r_2^2 - r_1^2) d\theta$.

Example:

Calculating the area between a circle $r=3$ and a cardioid $r=2+\cos(\theta)$ requires using the Area between Two Polar Curves formula.

Area of a Polar Region (single curve)

Criticality: 2

The area enclosed by a single polar curve from an angle $a$ to an angle $b$, calculated using the integral $\frac{1}{2}\int_a^b r^2 d\theta$.

Example:

To find the area of a single cardioid $r = 1 + \cos(\theta)$ , you would use the Area of a Polar Region (single curve) formula from $0$ to $2\pi$ .

Bounds (for polar area)

Criticality: 3

Example:

To find the area of a loop of a limacon, you'd need to determine the bounds (for polar area) where the curve intersects the origin or itself.

Double Angle Theorem

Criticality: 2

Trigonometric identities that express trigonometric functions of twice an angle in terms of functions of the original angle, such as $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$.

Example:

When integrating $\sin^2(\theta)$ , you might use the Double Angle Theorem to rewrite it as $\frac{1-\cos(2\theta)}{2}$ for easier integration.

Inside Radius ($r_1$)

Criticality: 3

In the context of finding the area between two polar curves, $r_1$ refers to the function representing the curve that is closer to the origin in the region of interest.

Example:

When calculating the area between $r=5$ and $r=3$ , $r=3$ would be your inside radius ( $r_1$ ).

Outside Radius ($r_2$)

Criticality: 3

In the context of finding the area between two polar curves, $r_2$ refers to the function representing the curve that is farther from the origin in the region of interest.

Example:

If you're finding the area between $r=5$ and $r=3$ , then $r=5$ would be your outside radius ( $r_2$ ).

Polar Coordinate System

Criticality: 2

A two-dimensional coordinate system where each point is determined by a distance from a reference point (the origin) and an angle from a reference direction (the positive x-axis).

Example:

Plotting a point like $(5, \pi/4)$ involves using the Polar Coordinate System.

Polar Curves

Criticality: 3

Curves defined by equations in polar coordinates, where points are specified by a distance from the origin (radius) and an angle from the positive x-axis.

Example:

The graph of $r = 2\cos( heta)$ is a polar curve representing a circle.

Polar Interval (a and b)

Criticality: 3

The angular range, represented by $a$ and $b$, over which the area of a polar region or between polar curves is calculated. These values are typically in radians.

Example:

For a full circle, the polar interval might be from $0$ to $2\pi$ .

Radian Values

Criticality: 3

A unit of angle measurement, where one radian is the angle subtended at the center of a circle by an arc equal in length to the radius. Calculus typically uses radians for angles.

Example:

When evaluating $\sin(\theta)$ in calculus, $\theta$ is almost always assumed to be in radian values.

Radius (r)

Criticality: 3

In polar coordinates, $r$ represents the distance of a point from the origin. In polar equations, it is typically a function of the angle $\theta$.

Example:

In the polar equation $r = 4\sin(\theta)$ , the radius $r$ changes with the angle $\theta$ .

Second Quadrant

Criticality: 2

The region in the Cartesian coordinate system where x-coordinates are negative and y-coordinates are positive, corresponding to angles between $\pi/2$ and $\pi$ in polar coordinates.

Example:

A point with polar coordinates $(r, \theta)$ where $\theta = 2\pi/3$ would lie in the Second Quadrant.