Glossary
Acceleration
Acceleration is the rate of change of an object's velocity with respect to time, indicating how quickly its velocity is changing.
Example:
A rocket's acceleration might be a(t) = 9.8 m/s², meaning its velocity increases by 9.8 m/s every second due to gravity.
Arc Length
(BC Only) Arc length is the measure of the distance along a curved path of a function between two points, calculated using a specific integral formula involving the derivative of the function.
Example:
To determine the exact length of a parabolic cable supporting a suspension bridge, you would calculate its arc length over the span of the bridge.
Area Between Curves
The area between two curves is the region bounded by the graphs of two functions over a specified interval, calculated by integrating the absolute difference between the functions.
Example:
To find the area between curves y = x² and y = x, you would integrate (x - x²) from their intersection points.
Average Value of a Function
The average value of a function over an interval represents the height of a rectangle with the same width as the interval and an area equal to the area under the function's curve over that interval.
Example:
To find the average value of a student's study effort (modeled by a function) over a 5-hour period, you would integrate the effort function and divide by 5.
Definite Integral
A definite integral calculates the accumulated change of a function over a specific interval, representing the net area under its curve.
Example:
Using a definite integral, you can calculate the total amount of water that has leaked from a tank over a given hour if you know the rate of leakage.
Disc Method
The disc method is used to find the volume of a solid of revolution where the cross-sections perpendicular to the axis of rotation are solid circular discs.
Example:
Using the disc method, you can calculate the volume of a sphere by revolving a semicircle around the x-axis.
Displacement
Displacement is the net change in an object's position from its starting point to its ending point, found by integrating the velocity function.
Example:
If a runner moves forward 10 meters and then backward 5 meters, their displacement is 5 meters from the start, regardless of the total distance covered.
Position
In particle motion, position describes the location of an object at a specific moment in time.
Example:
If a particle's position is given by s(t) = t^2 - 4t, then at t=1 second, its position is s(1) = -3 units.
Solid of Revolution
A solid of revolution is a three-dimensional shape formed by rotating a two-dimensional region around an axis.
Example:
Rotating the region under y = √x from x=0 to x=4 about the x-axis creates a solid of revolution resembling a paraboloid.
Total Distance Traveled
Total distance traveled is the sum of the magnitudes of all movements an object makes, regardless of direction, found by integrating the speed (absolute value of velocity) function.
Example:
If a car drives 5 miles east and then 3 miles west, the total distance traveled is 8 miles.
Velocity
Velocity is the rate of change of an object's position with respect to time, indicating both its speed and direction.
Example:
If a car's velocity is v(t) = 2t + 3 m/s, it means its speed is increasing over time.
Volume by Cross Sections
Volume by cross sections is a method to find the volume of a 3D solid by integrating the area of its 2D cross-sections perpendicular to an axis.
Example:
You can find the volume by cross sections of a solid whose base is a circle and whose cross-sections perpendicular to the x-axis are squares.
Washer Method
The washer method is used to find the volume of a solid of revolution when the cross-sections perpendicular to the axis of rotation are ring-shaped (washers) with an inner and outer radius.
Example:
To find the volume of a donut-shaped object (a torus), you would use the washer method by revolving a circle around an axis that does not pass through its center.