# Area, Volume, and Arc Length

### Terms

### Area Under The Curve

The area of the region bounded by the graph of function*f*(

*x*) and the x-axis for a ≤ x ≤ b, is given by the definite integral .

### Polar Coordinates

Polar coordinates use a two-dimensional coordinate system where each point is represented by a distance and angle from a fixed point.### Volumes Of Solids Of Revolution

The volume of the solid figure obtained by rotating a curve around a straight line.### Disk Method

This is a method for finding the volumes of solids of revolution when the cross sections are taken perpendicular to the axis of revolution and the axis is the boundary of the plane region.### Washer Method

This is a method for finding the volumes of solids of revolution when the cross sections are taken perpendicular to the axis of revolution and the axis is*not*the boundary of the plane region.

### Arc Length

The arc-length of a continuous function*f*(

*x*) on a closed interval [a,b] is defined as .

### Parametric Equations

The equations in which rather than the coordinates*x*and

*y*being defined in terms of each other, they are defined in terms of another parameter

*t*.

### Improper Integral

A definite integral is called an improper integral when the limits of integration are infinite (*a*=-∞,

*b*=∞) or the function becomes unbounded in [

*a*,

*b*].

### Reimann Sum

The Reimann Sum of a function*f*(

*x*) over an interval [

*a*,

*b*] is defined as where

*x*and

_{i-1 }< x_{i}^{* }< x_{i},*x*for

_{i}*i*= 1,…,

*n*divides the interval [

*a*,

*b*] into

*n*subintervals.

### Left-Hand Sum

The Left-Hand Sum of a function*f*(

*x*) over an interval [

*a*,

*b*] is defined as where

*x*for

_{i}*i*= 1,…,

*n*divides the interval [

*a*,

*b*] into

*n*subintervals.

### Right-Hand Sum

The Right-Hand Sum of a function*f*(

*x*) over an interval [

*a*,

*b*] is defined as where

*x*for

_{i}*i*= 1,…,

*n*divides the interval [

*a*,

*b*] into

*n*subintervals.

### Midpoint Sum

The Midpoint Sum of a function*f*(

*x*) over an interval [

*a*,

*b*] is defined as where , and

*x*for

_{i}*i*= 1,…,

*n*divides the interval [

*a*,

*b*] into

*n*subintervals.

### Trapezoid Sum

The trapezoid sum is the average between the Left-Hand and Right-Hand Sum.### Average Value Of A Function

Average value of a continuous function f on the closed interval [*a*,

*b*] is defined as .

### Concavity

The concavity of a function describes whether the function is curving up, down or not curving at all.Advertisement

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