June 4, 2026 - Volumes of Solids of Revolution Lecture

Vocabulary and conceptual terms regarding the calculation of volumes for solids of revolution using the disc and washer methods.

Radius of a circle cross-section

The height from the x-axis to the function $f(x)$.

Volume of a solid of revolution

The integral from $a$ to $b$ of the area function, represented as the integral of $\pi [f(x)]^2 dx$.

Parabolic function

A function like $f(x) = x^2 - 4x + 5$ which produces a parabola-shaped graph.

Slicing method

A technique for calculating volume by summing the areas of infinitely many cross-sections.

Disc method

Another name for the slicing method, referring to the shape created when an approximated rectangle is spun around an axis.

Riemann sum aspect

The conceptual basis for the disc method, where volume is approximated by spinning rectangles used in area sums.

Horizontal distance

The value of $x$ when solving a function for $x$ in terms of $y$, used when revolving around the y-axis.

Delta $dy$

The thickness of a disc or washer when integrating with respect to the y-axis.

Vertex

The peak or base point of a parabola, such as the point near $x = 2$ for the graph of $x^2 - 4x + 5$.

Units cubed ($units^3$)

The generic measurement unit used for volume when specific dimensions like inches or feet are not given.

Washer

A cross-section with a hole in the middle, created by revolving the area between two functions.

Area of a washer

The area calculated by subtracting the inner circle's area from the outer circle's area: $\pi ([f(x)]^2 - [g(x)]^2)$..

Washer method

A method for finding the volume of a solid with a hollow center by considering the distance between an upper bound function and a lower bound function.

Intersection points

The points where two curves meet, which must be identified to determine integration bounds if they are not provided.

Non-negative condition

The requirement that functions $f(x)$ and $g(x)$ must be zero or positive for the theorem on revolving regions to apply.

Radius shift

An adjustment made when revolving around a line other than the axis, such as adding $2$ to the function when revolving around $y = -2$.

Improper integrals

A type of integral with infinite bounds mentioned as the method for calculating infinite volume.

Chain rule

A calculus rule mentioned in the context of integrating squared polynomials, though foiling the expansion is often preferred.

Volume shells

An alternative method for finding the volume of solids of revolution mentioned as a separate technique from discs and washers.

Surface area

A measurement related to the exterior of the solid of revolution, noted as a future topic after volumes.