| Term | Definition |
|---|---|
| average value of a function | The mean value of a function over a specified interval, calculated using the formula (1/(b-a)) ∫[a to b] f(x) dx. |
| continuous | A function that has no breaks, jumps, or holes in its graph over a given interval. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| interval | A connected set of real numbers, typically expressed as a range between two endpoints. |
| Term | Definition |
|---|---|
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| disc method | A technique for finding the volume of a solid of revolution by integrating the cross-sectional areas of circular discs perpendicular to the axis of rotation. |
| solids of revolution | Three-dimensional solids formed by rotating a two-dimensional region around an axis. |
| Term | Definition |
|---|---|
| cross section | Two-dimensional slices of a three-dimensional solid, perpendicular to an axis, used to build up the volume through integration. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| ring shaped | Cross sections that have the shape of a washer or annulus, with an outer radius and an inner radius, used in volume calculations. |
| solids of revolution | Three-dimensional solids formed by rotating a two-dimensional region around an axis. |
| washer method | A technique for finding the volume of a solid of revolution by integrating the areas of ring-shaped (washer-shaped) cross sections perpendicular to the axis of rotation. |
| Term | Definition |
|---|---|
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| ring-shaped cross sections | Annular (donut-shaped) slices of a solid of revolution formed when the region being rotated has a gap between the axis of rotation and the outer boundary. |
| solids of revolution | Three-dimensional solids formed by rotating a two-dimensional region around an axis. |
| washer method | A technique for finding the volume of a solid of revolution by integrating the areas of ring-shaped (washer-shaped) cross sections perpendicular to the axis of rotation. |
| Term | Definition |
|---|---|
| arc length | The distance along a curve between two points, calculated using a definite integral. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| planar curve | A curve that exists in a two-dimensional plane and can be defined by a function or parametric equations. |
| Term | Definition |
|---|---|
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| displacement | The net change in position of a particle over a time interval, found by integrating the velocity vector. |
| rectilinear motion | Motion of a particle along a straight line, characterized by changes in position, velocity, and acceleration. |
| speed | The magnitude of the velocity vector, representing the rate at which a particle is moving without regard to direction. |
| total distance traveled | The total length of the path traveled by a particle over a time interval, found by integrating the speed. |
| velocity | The derivative of a position function with respect to time, representing the rate and direction of change of position for a moving particle. |
| Term | Definition |
|---|---|
| accumulation | The process of gathering or building up a quantity over time or over an interval, which can be expressed and calculated using definite integrals. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| net change | The total change in a quantity over an interval, calculated as the difference between final and initial values, often found using definite integrals. |
| rate of change | The measure of how quickly a quantity changes with respect to another variable, often time. |
| Term | Definition |
|---|---|
| areas in the plane | Regions bounded by curves and axes in a coordinate system whose measurements can be determined using integration. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| Term | Definition |
|---|---|
| areas in the plane | Regions bounded by curves and axes in a coordinate system whose measurements can be determined using integration. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| Term | Definition |
|---|---|
| absolute value of the difference | The absolute value of the difference between two functions, used to calculate area between curves regardless of which function is on top. |
| area between curves | The region enclosed between two or more curves, calculated using definite integrals. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| Term | Definition |
|---|---|
| cross section | Two-dimensional slices of a three-dimensional solid, perpendicular to an axis, used to build up the volume through integration. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| rectangular cross sections | Two-dimensional rectangular slices of a solid whose areas can be integrated to find the total volume. |
| square cross sections | Two-dimensional square slices of a solid whose areas can be integrated to find the total volume. |
| volumes of solids | The measure of three-dimensional space occupied by a solid object, calculated using integration techniques. |
| Term | Definition |
|---|---|
| area formulas | Mathematical expressions used to calculate the area of two-dimensional shapes, which are applied to cross sections in volume calculations. |
| cross section | Two-dimensional slices of a three-dimensional solid, perpendicular to an axis, used to build up the volume through integration. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| semicircular cross sections | Three-dimensional solids whose perpendicular slices are semicircular in shape. |
| triangular cross sections | Three-dimensional solids whose perpendicular slices are triangular in shape. |
| volumes of solids | The measure of three-dimensional space occupied by a solid object, calculated using integration techniques. |
| Term | Definition |
|---|---|
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| disc method | A technique for finding the volume of a solid of revolution by integrating the cross-sectional areas of circular discs perpendicular to the axis of rotation. |
| solids of revolution | Three-dimensional solids formed by rotating a two-dimensional region around an axis. |