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.

$\frac{1}{b-a} \int_{a}^{b} f(x) , dx$

$\int_{a}^{b} v(t) , dt$

$\int_{a}^{b} |v(t)| , dt$

$\int_{a}^{b} [f(x) - g(x)] , dx$

$\int [f(x)]^2 , dx$

$\int f(x) * w , dx$

$\int (1/2)*f(x)*w , dx$

$\int (1/2)[object Object](f(x))^2*w , dx$

$\int \pi [f(x)]^2 , dx$

$\int \pi (R^2 - r^2) , dx$

$\int_{a}^{b} \sqrt{1 + [f'(x)]^2} , dx$

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.