Volumes of Solids Using Slicing, Disk, and Washer Methods

Volumes by Slicing and Cross Sections

Introduction to Volume by Slicing

Calculating the volume of a solid can be accomplished by slicing the solid into thin pieces and summing their volumes. This approach is fundamental in calculus and is used to find volumes of irregular solids by integrating the area of cross sections.

• Key Concept: The volume of a solid with integrable cross-sectional area A(x) from x = a to x = b is given by the integral:

$V = \int_a^b A(x)\,dx$

• Cross Section: The cross section at position x is a slice perpendicular to the axis, whose area is A(x).

• Volume Approximation: The volume can be approximated by summing the volumes of many thin slices (e.g., cylinders or disks).

Steps for Calculating Volumes by Cross Sections

1. Sketch the solid and a typical cross section.

2. Find a formula for the area of a typical slice, A(x) or A(y).

3. Determine the bounds of integration.

4. Integrate $\int A(x)\,dx$ or $\int A(y)\,dy$ to find the volume.

Example: Volume of a Right Circular Cylinder

• Area of cross section: $A(x) = \pi r^2$ (where r is the radius)

• Volume: $V = \int_0^h \pi r^2\,dx = \pi r^2 h$

Example: Volume of a Solid with Square Cross Sections

• Base: Region between the curve $y = f(x)$ and the x-axis, from $x = a$ to $x = b$.

• Cross sections: Perpendicular to the x-axis, each is a square with base from x-axis to $y = f(x)$.

• Area: $A(x) = [f(x)]^2$

• Volume: $V = \int_a^b [f(x)]^2\,dx$

Example: Equilateral Triangle Cross Sections

• Area of triangle: If base is $b$, area is $A = \frac{\sqrt{3}}{4} b^2$

• Volume: $V = \int_a^b \frac{\sqrt{3}}{4} [f(x)]^2\,dx$

Volumes by Rotation: Disk and Washer Methods

Introduction to Solids of Revolution

When a plane region is rotated about an axis, it generates a three-dimensional solid. The volume of such solids can be calculated using the disk or washer methods, depending on whether the region is bounded by one or two curves.

• Disk Method: Used when the region is bounded by a single curve and the axis.

• Washer Method: Used when the region is bounded by two curves, creating a 'hole' in the solid.

Disk Method

• Volume of a thin disk: $dV = \pi [R(x)]^2 dx$

• Total volume: $V = \int_a^b \pi [R(x)]^2 dx$

• Example: Rotating $y = f(x)$ about the x-axis from $x = a$ to $x = b$:

$V = \int_a^b \pi [f(x)]^2 dx$

Washer Method

• Volume of a thin washer: $dV = \pi \left( [R(x)]^2 - [r(x)]^2 \right) dx$

• Total volume: $V = \int_a^b \pi \left( [R(x)]^2 - [r(x)]^2 \right) dx$

• Where: $R(x)$ is the outer radius, $r(x)$ is the inner radius.

Example: Volume of a Sphere

• Circle equation: $x^2 + y^2 = a^2$

• Rotating about x-axis: $y = \sqrt{a^2 - x^2}$

• Volume: $V = \int_{-a}^a \pi [\sqrt{a^2 - x^2}]^2 dx = \int_{-a}^a \pi (a^2 - x^2) dx$

• Result: $V = \frac{4}{3} \pi a^3$

Example: Volume by Rotation About the y-axis

• Region bounded by $x = f(y)$ and $x = g(y)$ from $y = c$ to $y = d$.

• Volume: $V = \int_c^d \pi \left( [R(y)]^2 - [r(y)]^2 \right) dy$

Summary Table: Methods for Calculating Volumes

Method	Formula	When to Use
Cross Section (Slicing)	$V = \int_a^b A(x) dx$	Solid with known cross-sectional area
Disk Method	$V = \int_a^b \pi [R(x)]^2 dx$	Region rotated about axis, no hole
Washer Method	$V = \int_a^b \pi \left( [R(x)]^2 - [r(x)]^2 \right) dx$	Region rotated about axis, with hole

Additional Info

• Volumes can also be calculated with respect to y if the cross sections are perpendicular to the y-axis.

• For solids with more complex cross sections (e.g., equilateral triangles, rectangles), the area formula for the shape is used in the integral.

• When rotating about lines other than the axes, adjust the radius formulas accordingly.