Glossary
Axis of Rotation
The fixed line around which a two-dimensional region is revolved to generate a three-dimensional solid of revolution.
Example:
For a sphere, the axis of rotation could be any diameter of the circle that generates it.
Boundaries/Endpoints
The specific values (a, b for x-axis; c, d for y-axis) that define the interval over which the region is revolved and the definite integral is evaluated.
Example:
When rotating the region from to , these values are the boundaries/endpoints of integration.
Definite Integral
An integral with upper and lower limits, representing the net accumulation or area under a curve over a specific interval, used here to sum the volumes of infinitesimal discs.
Example:
Calculating the total distance traveled by a car given its velocity function over a time interval involves evaluating a definite integral.
Disc Method
A specific technique for calculating the volume of a solid of revolution by summing the volumes of infinitesimally thin cylindrical discs perpendicular to the axis of rotation.
Example:
When finding the volume of a cone, you can use the disc method by stacking circular cross-sections.
Disc Method: X-Axis
The application of the disc method where the region is revolved around the x-axis, leading to an integral of the form $\int_{a}^{b}\pi (f(x))^2dx$.
Example:
To find the volume of a solid formed by rotating from to around the x-axis, you'd use the Disc Method: X-Axis.
Disc Method: Y-Axis
The application of the disc method where the region is revolved around the y-axis, requiring the function to be expressed in terms of y, leading to an integral of the form $\int_{c}^{d}\pi (f(y))^2dy$.
Example:
If you rotate the region bounded by and the y-axis from to around the y-axis, you'll apply the Disc Method: Y-Axis.
Radius of Disc
The distance from the axis of rotation to the outer edge of an infinitesimally thin disc, typically represented by the function $f(x)$ or $f(y)$.
Example:
If you're rotating around the x-axis, the radius of disc at any point x is simply .
Solids of Revolution
Three-dimensional shapes created by rotating a two-dimensional region around a fixed line, known as the axis of revolution.
Example:
A donut is a solid of revolution formed by rotating a circle around an axis outside the circle.
Volume with Disc Method
A calculus technique used to find the volume of a three-dimensional solid formed by revolving a two-dimensional region around an axis.
Example:
Imagine calculating the volume with disc method to find how much water a wine glass can hold if its shape is generated by rotating a curve.
Width of Disc
The infinitesimal thickness of each disc, denoted as $dx$ when revolving around the x-axis or $dy$ when revolving around the y-axis.
Example:
In the integral , the width of disc is represented by .