Glossary
Accumulation of Change
The fundamental concept that a definite integral sums up infinitesimal changes of a quantity over an interval to determine the total change or total amount.
Example:
When applying the Disc Method, we are performing an accumulation of change by summing the volumes of infinitely many thin discs to find the total volume of the solid.
Area
The two-dimensional region bounded by curves and lines that is rotated around an axis to generate a solid of revolution.
Example:
To calculate the volume of a wine glass, you would rotate the area defined by its profile curve and the y-axis.
Axis of Revolution
The straight line about which a two-dimensional region or curve is rotated to generate a three-dimensional solid of revolution.
Example:
When a potter spins clay on a wheel to form a vase, the central spindle of the wheel acts as the Axis of Revolution.
Axis of Rotation
The line around which a two-dimensional region is revolved to generate a three-dimensional solid.
Example:
If you spin a rectangle around its bottom edge, that edge acts as the Axis of Rotation, forming a cylinder.
Axis of Rotation
The fixed line around which a two-dimensional region is revolved to create a three-dimensional solid of revolution.
Example:
When making a spinning top on a lathe, the central rod acts as the Axis of Rotation.
Boundaries
The specific limits or endpoints of an interval over which an integral is calculated, typically determined by the intersection points of the functions defining the region.
Example:
When finding the area between two curves, the x-values where the curves cross define the Boundaries of integration.
Boundaries
The specific limits or values (e.g., y=a to y=b) that define the extent of the 2D region being rotated, which become the limits of integration in the volume formula.
Example:
When calculating the volume of a vase, the boundaries for integration might be from y=0 (the base) to y=10 (the top opening).
Bounds of Integration
The upper and lower limits of the independent variable over which a definite integral is evaluated, defining the specific interval for accumulation.
Example:
For the integral , the numbers 0 and 5 are the Bounds of Integration, indicating the interval over which the area is calculated.
Continuous function
A function whose graph can be drawn without lifting the pen, meaning it has no breaks, jumps, or holes over its domain.
Example:
For the Disc Method to be directly applicable, the function f(x) defining the curve must be a continuous function over the entire interval of integration.
Cross Section
A two-dimensional shape that results from slicing a three-dimensional object with a plane, often perpendicular to an axis of symmetry or rotation.
Example:
If you cut a cylindrical log straight across, the resulting circular face is a Cross Section of the log.
Cross-section
The shape formed by slicing a three-dimensional solid with a plane perpendicular to an axis. In volume calculations, these are often disks or washers.
Example:
If you slice a loaf of bread, each slice represents a Cross-section of the loaf.
Definite Integral
An integral with specified upper and lower limits, representing the net accumulation of a quantity or the signed area under a curve between those limits.
Example:
To find the total distance traveled by a car with a varying velocity, you would evaluate the Definite Integral of its velocity function over a given time interval.
Definite Integral
An integral with specified upper and lower limits, representing the net accumulation of a quantity over a given interval and yielding a single numerical value.
Example:
Calculating the definite integral of x² from 0 to 1 gives the exact area under the curve y = x² between those points, which is a foundational step in volume calculations.
Definite Integral
A mathematical tool used to calculate the net accumulation of a quantity over a specific interval, often representing area under a curve or volume.
Example:
Calculating the total distance traveled by a car given its velocity function over time involves evaluating a Definite Integral.
Disc Method
A technique in calculus used to find the volume of a solid of revolution by summing the volumes of infinitesimally thin, circular discs perpendicular to the axis of rotation.
Example:
To calculate the volume of a sphere, one can imagine slicing it into many thin circular discs and using the Disc Method to sum their volumes.
Disc Method
A technique for calculating the volume of a solid of revolution by summing the volumes of infinitesimally thin cylindrical discs. It is applied when the region being rotated is adjacent to the axis of revolution.
Example:
To find the volume of a sphere, one could use the Disc Method by rotating a semicircle around the x-axis, treating it as a stack of countless thin circular discs.
Disc Method
A calculus technique used to find the volume of a solid of revolution by summing the volumes of infinitesimally thin cylindrical discs perpendicular to the axis of rotation.
Example:
Using the Disc Method, you can determine the volume of a sphere by imagining it as a stack of countless thin circular slices, each with a tiny thickness.
Disk Method
A special case of the washer method used when the region being rotated is flush against the axis of rotation, resulting in a solid with no inner hollow.
Example:
To find the volume of a cone, you could rotate a right triangle around one of its legs using the Disk Method.
Function (Rewriting)
The algebraic process of expressing a function in terms of the variable of integration (e.g., converting y = f(x) to x = g(y) when revolving around the y-axis).
Example:
If you are given y = x^3 and need to revolve it around the y-axis, you must perform function rewriting to get x = y^(1/3) before setting up the integral.
Functions (of y)
Equations expressed in the form $x = f(y)$ or $x = g(y)$, which define the outer and inner boundaries of the 2D region being rotated around the y-axis.
Example:
When revolving around the y-axis, you need to express your curves as functions (of y), such as or .
Gap
In the context of the washer method, a gap refers to the empty space between the region being rotated and the axis of rotation, which results in a hollow solid.
Example:
If you rotate the region between y = x² + 1 and y = 5 around the x-axis, there's a gap between the x-axis and the lower curve y = x² + 1, necessitating the washer method.
Inner Radius
In the washer method, this is the distance from the axis of rotation to the inner boundary of the rotated region, typically represented by the function closer to the axis.
Example:
When rotating the region between y = x and y = x² around the x-axis, y = x² acts as the Inner Radius for x values between 0 and 1.
Integral
A fundamental concept in calculus representing the accumulation of quantities, often used to find areas, volumes, or total change.
Example:
To find the total distance traveled by a car given its speed over time, you would compute the Integral of the speed function.
Integral
A fundamental concept in calculus used to find the total accumulation of a quantity, such as area under a curve or volume of a solid.
Example:
To determine the exact volume using the washer method, you must set up and solve a definite integral.
Integral
A fundamental concept in calculus used to find the total accumulation of a quantity, such as area, volume, or total change, by summing infinitesimal parts.
Example:
To find the total amount of water that flows into a tank over time, given a flow rate function, you would use an integral.
Integrand
The function or expression that is being integrated in a definite or indefinite integral.
Example:
In the integral , the expression is the Integrand.
Integrand
The function or expression that is being integrated within an integral.
Example:
In the volume formula V = π ∫[a to b] [f(x)]² dx, the term [f(x)]² is the integrand, representing the area of a single circular cross-section.
Integration Interval/Boundaries
The specific range of y-values, denoted as $[a, b]$, over which the definite integral is evaluated to calculate the total volume.
Example:
If the region extends from to , then represents the integration interval for your volume calculation.
Limits of Integration
The upper and lower bounds (a and b) of a definite integral, which specify the interval over which the integration is performed.
Example:
In the volume formula V = π ∫[a to b] [f(x)]² dx, a and b are the limits of integration that define the starting and ending points of the rotation along the x-axis.
Outer Radius
In the washer method, this is the distance from the axis of rotation to the outer boundary of the rotated region, typically represented by the function further from the axis.
Example:
When rotating the region between y = x and y = x² around the x-axis, y = x acts as the Outer Radius for x values between 0 and 1.
Points of Intersection
The coordinates where two or more curves meet, which are crucial for determining the limits of integration (boundaries) in volume calculations.
Example:
To set up the integral for the volume of a solid formed by rotating the region between y=x and y=x^2, you first find their Points of Intersection at (0,0) and (1,1).
Region (of Revolution)
The two-dimensional area, bounded by curves and lines, that is rotated around an axis to generate a three-dimensional solid of revolution.
Example:
For a problem asking to find the volume of a solid formed by rotating y = x² from x = 0 to x = 2 around the x-axis, the region is the specific area under the parabola between those x-values.
Solid of Revolution
A three-dimensional geometric shape formed by rotating a two-dimensional curve or region around a straight line (the axis of revolution) in three-dimensional space.
Example:
A wine glass, when viewed from the side, can be seen as a Solid of Revolution formed by rotating a specific curve around a vertical axis.
Solid of Revolution
The three-dimensional object created when a two-dimensional region is rotated around an axis.
Example:
Rotating a semi-circle around its diameter creates a sphere, which is a Solid of Revolution.
Solid of Revolution
A three-dimensional shape created by rotating a two-dimensional region around a fixed line, known as the axis of revolution.
Example:
Rotating the area under the curve y = x² from x = 0 to x = 2 around the x-axis forms a solid of revolution that resembles a bell shape.
Solid of Revolution
A three-dimensional shape created by rotating a two-dimensional planar region around a straight line, known as the axis of revolution.
Example:
If you rotate a rectangle around one of its sides, the resulting solid of revolution is a perfect cylinder.
Solid of Revolution
A three-dimensional shape formed by rotating a two-dimensional region around a line, known as the axis of rotation.
Example:
Rotating a semi-circle around its diameter creates a Solid of Revolution in the shape of a sphere.
Volume of Revolution
The three-dimensional space occupied by a solid created when a two-dimensional region in a plane is rotated around a specified line, known as the axis of revolution.
Example:
If you rotate a semi-circle around its diameter, the resulting Volume of Revolution is a sphere.
Volume of Revolution
The volume of a three-dimensional solid generated by rotating a two-dimensional region around a given axis.
Example:
Imagine spinning a flat, triangular piece of cardboard around one of its sides; the resulting 3D shape, like a cone, has a specific Volume of Revolution.
Volume of Revolution
The three-dimensional volume generated when a two-dimensional region is rotated around a specific axis.
Example:
Rotating a rectangle around one of its sides creates a cylinder, which is a simple Volume of Revolution.
Volume of Revolution
The measure of the space occupied by a solid of revolution, typically calculated using integration techniques like the Disc Method.
Example:
If you rotate the region bounded by y = x and the x-axis from x = 0 to x = 1, the resulting volume of revolution would be the volume of a cone.
Volume of Revolution
The volume of a three-dimensional solid formed by rotating a two-dimensional region around a specific axis.
Example:
When you spin a flat, crescent-shaped region around the y-axis, the resulting 3D shape's total space occupied is its volume of revolution.
Volume of Revolution
The measure of the three-dimensional space occupied by a solid generated by revolving a planar region around a fixed line.
Example:
The Volume of Revolution of a rectangle rotated around one of its sides is a cylinder.
Washer Method
A calculus technique used to find the volume of a solid of revolution that has a hole, by integrating the difference between the areas of two concentric circles (washers).
Example:
When calculating the volume of a donut-shaped object, the Washer Method allows you to subtract the volume of the inner hole from the total volume.
Washer Method
A technique used to calculate the volume of a solid of revolution, particularly when there is a gap between the region being rotated and the axis of revolution.
Example:
To find the volume of a solid shaped like a ring or a hollow cylinder, you would typically apply the Washer Method.
Washer Method
A technique used to calculate the volume of a solid of revolution when there is a hollow space or gap between the rotated region and the axis of rotation.
Example:
When rotating the area between y = x and y = x² around the x-axis, you'd use the Washer Method to find the volume of the resulting hollow shape.
x-axis (as axis of rotation)
The horizontal axis in a Cartesian coordinate system around which a two-dimensional region is revolved to form a three-dimensional solid.
Example:
When rotating the region bounded by y = sin(x) and the x-axis from x = 0 to x = π, the resulting solid will be symmetrical around the x-axis.
y-axis (Axis of Revolution)
The vertical line around which a two-dimensional region is rotated to generate a three-dimensional solid, requiring functions to be expressed in terms of y.
Example:
When a region is rotated around the y-axis, the resulting solid will have circular cross-sections perpendicular to the y-axis.