The value a function would take at a single point if the area under the curve equaled the area of a rectangle with the same width and height.

A vector quantity representing the change in position of an object.

A scalar quantity representing the total distance covered by an object, regardless of its final position.

A three-dimensional shape formed by rotating a two-dimensional region about an axis.

A measure of the distance along the curved path of a function.

The rate of change (derivative) of the position as a function of time.

The rate of change (derivative) of velocity as a function of time.

A method to find the volume of a solid of revolution by cutting it into thin disks.

A method to find the volume of a solid of revolution using ring-shaped objects (washers).

The intersection of a solid with a plane.

1. Find the definite integral of f(x) from a to b. 2. Divide the result by (b-a).

1. Integrate v(t) from a to b.

1. Find intervals where v(t) is positive and negative. 2. Integrate |v(t)| over [a, b], summing the absolute values of the integrals over each interval.

1. Determine which function is greater. 2. Integrate the difference between the greater and lesser function from a to b.

1. Identify the radius f(x). 2. Integrate $\pi [f(x)]^2$ over the interval.

1. Identify the outer radius R and inner radius r. 2. Integrate $\pi (R^2 - r^2)$ over the interval.

1. Determine the side length of the square, f(x). 2. Integrate $[f(x)]^2$ over the interval.

1. Find the derivative f'(x). 2. Square the derivative. 3. Add 1. 4. Take the square root. 5. Integrate from a to b.

1. Find the points of intersection. 2. Divide the interval into subintervals. 3. Integrate the absolute value of the difference between the functions over each subinterval. 4. Sum the results.

1. Express the area of the cross-section as a function of x, A(x). 2. Integrate A(x) with respect to x over the interval [a, b].

Displacement: Change in position, can be negative. | Total Distance: Total path length, always non-negative.

Disc: Solid of revolution has no hole. | Washer: Solid of revolution has a hole.

x: Use vertical rectangles, integrate along x-axis. | y: Use horizontal rectangles, integrate along y-axis.

Functions of x: Integrate (top - bottom) with respect to x. | Functions of y: Integrate (right - left) with respect to y.

Slicing: General method for any cross-sectional shape. | Disc/Washer: Specific to solids of revolution with circular/ring cross-sections.

Area: Integrates a function representing height to find 2D space. | Volume: Integrates a function representing area to find 3D space.

Squares: Volume is integral of (side length)^2. | Circles: Volume is integral of $\pi * (radius)^2$.

Disc: Radius is the distance from the function to the axis. | Washer: Requires both inner and outer radii relative to the axis.

Net Change: Direct integration gives the difference between endpoints. | Total Accumulation: Requires considering absolute values or intervals of increase/decrease.

Physics: Used for motion, work, energy. | Economics: Used for cost, revenue, consumer surplus.