Glossary
Bounds of Integration
The specific x or y values that define the interval over which the integral is evaluated, typically representing the intersection points of the functions that form the region.
Example:
Finding where the two functions intersect will give you the bounds of integration, which are crucial for setting up your definite integral correctly.
Horizontal Line of Revolution (y = b)
A horizontal line, represented by the equation y = b, around which a 2D region is revolved to generate a 3D solid, requiring adjustments to the radius formulas.
Example:
If your solid is spun around the line y = -2, this acts as your horizontal line of revolution, affecting how you define your radii.
Inner Radius (g(x))
The distance from the axis of revolution to the inner boundary of the region being revolved, representing the smaller radius that forms the hole in the washer method.
Example:
The function g(x) that is closer to the axis of revolution determines the inner radius, which is subtracted to account for the hollow part of the solid.
Outer Radius (f(x))
The distance from the axis of revolution to the outer boundary of the region being revolved, representing the larger radius in the washer method's integral.
Example:
In the integral, the function f(x) that is farther from the axis of revolution defines the outer radius of each washer.
Revolving Around Other Axes
The process of rotating a 2D region around a horizontal line (y=b) or a vertical line (x=a) that is not the x-axis or y-axis, respectively, to form a 3D solid.
Example:
Instead of the x-axis, you might be asked to find the volume by revolving around other axes, such as the line y = -2, which shifts the center of rotation.
Vertical Line of Revolution (x = a)
A vertical line, represented by the equation x = a, around which a 2D region is revolved to generate a 3D solid, typically requiring integration with respect to y.
Example:
When calculating the volume of a solid spun around x = 3, this vertical line of revolution means your radii will be expressed in terms of y.
Volume with Washer Method
A calculus technique used to find the volume of a solid of revolution when the region being revolved has a hole in the middle, formed by revolving an area between two curves.
Example:
To calculate the volume with washer method for a solid shaped like a ring, you'd integrate the difference of the squared outer and inner radii.
Washer Method
A technique for calculating the volume of a solid of revolution by integrating the difference between the areas of two concentric circles (an outer radius and an inner radius).
Example:
The washer method is perfect for finding the volume of a solid that resembles a donut or a pipe, where there's a clear hole in the center.