Glossary
Accumulation of Change
The fundamental concept behind integration, where infinitesimal changes of a quantity are summed up over an interval to determine the total change or total amount of that quantity.
Example:
The accumulation of change in a company's profit rate over a fiscal year reveals its total profit for that period.
Area of the Cross Section
The area of a two-dimensional slice of a three-dimensional solid, taken perpendicular to a given axis. This area is often expressed as a function, A(x) or A(h).
Example:
When slicing a cucumber, the circular face you see is the area of the cross section at that particular cut.
Base (of a triangle)
Any side of a triangle that is chosen as the reference for calculating its area, from which the height is measured perpendicularly.
Example:
When calculating the area of a triangular prism, the area of its base triangle is a key component.
Continuous Function
A function whose graph can be drawn without lifting the pen, meaning it has no breaks, jumps, or holes in its domain.
Example:
The temperature change over a day can be modeled by a continuous function, as it doesn't suddenly jump from one value to another.
Continuous Function
A function whose graph can be drawn without lifting the pen, meaning there are no breaks, jumps, or holes in its domain.
Example:
The temperature change over a day is typically a Continuous Function, as it doesn't suddenly jump from one value to another without passing through intermediate temperatures.
Cross Section
A surface or shape exposed by making a straight cut through a solid. Its area is fundamental for calculating the solid's volume using integration.
Example:
If you slice a cucumber, the circular shape you see on the cut surface is a cross section.
Cross Section
A two-dimensional shape formed by the intersection of a plane with a three-dimensional solid. It's essentially a 'slice' of the object.
Example:
If you cut a loaf of bread, each slice represents a cross section of the loaf.
Cross-Sectional Area ($A(x)$)
The area of the shape obtained by cutting through a solid perpendicular to a given axis at position $x$. This function is integrated to find the volume of the solid.
Example:
When designing a bridge, engineers might calculate the Cross-Sectional Area of a support beam at different points to ensure its structural integrity.
Cross-Sectional Area Function ($A(x)$)
A function that describes the area of a 2D slice (cross-section) of a 3D solid at a particular value along an axis, typically $x$.
Example:
If a solid has a circular base and its cross-sections perpendicular to the x-axis are squares, the Cross-Sectional Area Function would give the area of those squares at each x-coordinate.
Cross-section
A surface or shape exposed by making a straight cut through a three-dimensional object, typically perpendicular to a given axis.
Example:
When slicing a cucumber, each slice reveals a circular cross-section of the vegetable.
Cross-sectional Area Function
A function, typically denoted as A(x) or A(y), that describes the area of a solid's cross-section at a specific point along an axis.
Example:
If a solid has square cross-sections whose side length is given by f(x), then the cross-sectional area function would be A(x) = [f(x)]^2.
Cross-sectional Area Function (A(x))
A function that describes the area of a solid's cross section at a given point along an axis, typically 'x' or 'y'. This function is integrated to find the total volume.
Example:
For a solid whose cross sections are squares with side length , the cross-sectional area function would be .
Definite Integral
A mathematical operation that calculates the net accumulation of a quantity over a specified interval. It represents the exact area under a curve or the total quantity accumulated between two points.
Example:
To find the total displacement of a particle given its velocity function over a time interval, you would use a definite integral.
Diameter
The distance across a circle or sphere passing directly through its center, which is always twice the length of the radius.
Example:
If a car tire has a Diameter of 60 cm, then its radius is 30 cm.
Equilateral Triangle
A triangle in which all three sides are equal in length, and consequently, all three interior angles are also equal (each 60 degrees).
Example:
The famous 'impossible triangle' optical illusion is often depicted using the shape of an equilateral triangle.
Height (of a triangle)
The perpendicular distance from the chosen base of a triangle to its opposite vertex.
Example:
To find the area of a triangular flag, you need to measure its height from the bottom edge to the top corner.
Integral
A mathematical object that represents the accumulation of quantities, often used to find the area under a curve or the volume of a solid.
Example:
Using an integral, a physicist can calculate the total work done by a variable force over a certain distance.
Integral
A fundamental concept in calculus used to find the total accumulation of a quantity, such as the area under a curve or the volume of a solid.
Example:
Using an integral, you can calculate the total work done by a variable force over a certain distance.
Integral
A mathematical operation that represents the accumulation of quantities, often visualized as the area under a curve or the volume of a solid.
Example:
To calculate the total work done by a spring as it's stretched, you would compute the Integral of the force function over the displacement.
Integral
A fundamental concept in calculus used to find the total accumulation of a quantity, often representing the area under a curve or the volume of a solid by summing infinitesimal slices.
Example:
To determine the total amount of water flowing into a reservoir over a day, given its flow rate function, one would compute the Integral of the flow rate over time.
Integral Calculus
A branch of calculus concerned with the theory and applications of integrals, including finding areas, volumes, and other accumulations.
Example:
Integral calculus is essential for engineers designing structures, as it helps them calculate stresses and strains over complex surfaces.
Integral Calculus
A branch of calculus focused on the accumulation of quantities, primarily used to find areas, volumes, and other total sums by integrating functions.
Example:
When determining the total charge accumulated on a capacitor over time, given a varying current, you would apply principles of Integral Calculus.
Integrand
The function that is being integrated within an integral expression.
Example:
In the expression ∫(x^3 + 2x) dx, the function (x^3 + 2x) is the integrand.
Limits of Integration
The upper and lower bounds of an integral, which define the specific interval over which the integration is performed.
Example:
In the expression , the numbers 1 and 5 are the limits of integration, specifying the start and end points for the calculation.
Limits of Integration
The upper and lower bounds of an integral, which define the specific interval over which the integration is performed.
Example:
When calculating the area under a curve from x=1 to x=5, 1 and 5 are the limits of integration.
Limits of Integration
The upper and lower bounds of the interval over which a definite integral is evaluated. These values define the specific range for the accumulation or summation.
Example:
When calculating the total work done by a force from point A to point B, A and B serve as the limits of integration for the work integral.
Limits of Integration
The upper and lower bounds of the interval over which an integral is calculated, defining the specific range for accumulation.
Example:
When finding the volume of a solid from to , the values 0 and 5 are the Limits of Integration.
Radius
The distance from the center of a circle or sphere to any point on its circumference or surface.
Example:
A compass set to draw a circle with a 5 cm opening is effectively setting the Radius of the circle to 5 cm.
Semicircle
A two-dimensional geometric shape that constitutes exactly half of a circle, bounded by a diameter and half of the circumference.
Example:
If you slice a perfectly round cookie exactly in half, each piece forms a Semicircle.
Solid with Known Cross Sections
A three-dimensional object whose volume can be determined by integrating the areas of its known cross-sectional shapes along an axis.
Example:
A pyramid can be modeled as a solid with known cross sections (squares) that decrease in size from base to apex.
Volume
The amount of three-dimensional space occupied by an object. In calculus, it's often calculated by integrating the area of cross sections.
Example:
To determine how much water a oddly shaped tank can hold, you would calculate its volume.
Volume
The amount of three-dimensional space occupied by a solid. In calculus, it is often found by integrating the cross-sectional area of the solid.
Example:
To determine how much water a uniquely shaped tank can hold, you would calculate its volume using integral calculus.
Volume
The amount of three-dimensional space occupied by a solid object, typically measured in cubic units. In calculus, it's often found by integrating cross-sectional areas.
Example:
Calculating the volume of a complex architectural structure helps engineers determine the amount of material needed for construction.
Volumes with Rectangular Cross Sections
A method in calculus used to compute the volume of a 3D solid by summing up infinitesimally thin rectangular slices perpendicular to a given axis.
Example:
To find the total amount of material needed to build a ramp with a varying rectangular profile, you would calculate its Volumes with Rectangular Cross Sections.