Lesson 24: Integrals in Mass Calculations (16.6)

Accessible transcription generated on 3/13/2026

Original Notes

Page 1

Lesson 24: Integrals in Mass Calculations (16.6)

Warmup:

At what point do the objects balance?

(1)

A schematic diagram labeled '(1)' illustrating a physics balance problem. A horizontal beam is balanced on a central triangular fulcrum. On the left end of the beam is a square weight labeled '10 LB'. On the right end is another square weight also labeled '10 LB'. The distance from the left weight to the fulcrum is denoted by a blue curly brace and the label 'x_1'. The distance from the right weight to the fulcrum is denoted by a similar blue curly brace and the label 'x_2'.
Visual Description: A schematic diagram labeled '(1)' illustrating a physics balance problem. A horizontal beam is balanced on a central triangular fulcrum. On the left end of the beam is a square weight labeled '10 LB'. On the right end is another square weight also labeled '10 LB'. The distance from the left weight to the fulcrum is denoted by a blue curly brace and the label 'x_1'. The distance from the right weight to the fulcrum is denoted by a similar blue curly brace and the label 'x_2'.
Guess: Center

(2)

A schematic diagram labeled '(2)' illustrating a second balance problem. A horizontal beam is supported by a triangular fulcrum. On the left end is a square weight labeled '10 LB' at a distance 'x_1' from the fulcrum (indicated by a blue brace). On the right end is a larger square weight labeled '20 LB' at a distance 'x_2' from the fulcrum (indicated by a blue brace). The diagram shows the beam in a balanced state with the fulcrum positioned closer to the heavier weight, making 'x_1' appear longer than 'x_2'.
Visual Description: A schematic diagram labeled '(2)' illustrating a second balance problem. A horizontal beam is supported by a triangular fulcrum. On the left end is a square weight labeled '10 LB' at a distance 'x_1' from the fulcrum (indicated by a blue brace). On the right end is a larger square weight labeled '20 LB' at a distance 'x_2' from the fulcrum (indicated by a blue brace). The diagram shows the beam in a balanced state with the fulcrum positioned closer to the heavier weight, making 'x_1' appear longer than 'x_2'.
Guess: towards heavier weight

Moment = distance from reference point \(\times\) Mass
Center of Mass = Point where moments are same

(1) \(x_1 \times 10 = x_2 \times 10 \rightarrow x_1 = x_2\)

(2) \(x_1 \times 10 = x_2 \times 20 \rightarrow x_1 = 2x_2\)

Page 2

Mass and Moment of a thin Plate

density = \(\rho(x,y)\)

A coordinate geometry diagram showing an irregularly shaped thin plate region, labeled D, situated in the first quadrant of an xy-plane. The region is filled with diagonal hatching. A small rectangular differential area element within the region is highlighted and labeled \(\Delta A\). A horizontal blue double-headed arrow, labeled 'x', indicates the distance from the y-axis to the center of the area element. A vertical green double-headed arrow, labeled 'y', indicates the distance from the x-axis to the same area element. This diagram illustrates the geometric basis for calculating mass and moments using density and coordinates.
Visual Description: A coordinate geometry diagram showing an irregularly shaped thin plate region, labeled D, situated in the first quadrant of an xy-plane. The region is filled with diagonal hatching. A small rectangular differential area element within the region is highlighted and labeled \(\Delta A\). A horizontal blue double-headed arrow, labeled 'x', indicates the distance from the y-axis to the center of the area element. A vertical green double-headed arrow, labeled 'y', indicates the distance from the x-axis to the same area element. This diagram illustrates the geometric basis for calculating mass and moments using density and coordinates.

Mass of area element \(\Delta A = \rho(x,y) \Delta A\)

Mass of \(D = M = \iint_D \rho(x,y) dA\)

\(M_y\) of \(\Delta A\) = Moment about \(y\)-axis
= distance from \(y\)-axis \(\times\) Mass
= \(x \cdot \rho(x,y) \Delta A\)

\[M_y \text{ for } D = \iint_D x \rho(x,y) dA\]

\(M_x\) of \(\Delta A\) = Moment about \(x\)-axis
= distance from \(x\)-axis \(\times\) Mass
= \(y \cdot \rho(x,y) \Delta A\)

\[M_x \text{ for } D = \iint_D y \rho(x,y) dA\]

Page 3

Center of Mass of a thin Plate

The center of mass is the point \( (\bar{x}, \bar{y}) \) where you can think the entire plate is a point mass.
A technical sketch on a 2D coordinate system (x and y axes). An irregularly shaped region, representing a thin plate with domain D, is drawn in the first quadrant and filled with diagonal hatching lines to indicate the distribution of mass.
Visual Description: A technical sketch on a 2D coordinate system (x and y axes). An irregularly shaped region, representing a thin plate with domain D, is drawn in the first quadrant and filled with diagonal hatching lines to indicate the distribution of mass.
A coordinate system diagram showing the concept of center of mass. A single point is plotted and labeled as (x-bar, y-bar), representing the concentration of the entire plate's mass. A blue horizontal bracket above the point indicates the distance x-bar from the y-axis, and a green vertical bracket to the right of the point indicates the distance y-bar from the x-axis.
Visual Description: A coordinate system diagram showing the concept of center of mass. A single point is plotted and labeled as (x-bar, y-bar), representing the concentration of the entire plate's mass. A blue horizontal bracket above the point indicates the distance x-bar from the y-axis, and a green vertical bracket to the right of the point indicates the distance y-bar from the x-axis.

\[ M_y = \iint_{D} x \rho(x, y) \, dA \quad \longleftrightarrow \quad \begin{matrix} M_y = \bar{x} M \\ \Rightarrow \bar{x} = \frac{1}{M} M_y \end{matrix} \]

\[ M_x = \iint_{D} y \rho(x, y) \, dA \quad \longleftrightarrow \quad \begin{matrix} M_x = \bar{y} M \\ \Rightarrow \bar{y} = \frac{1}{M} M_x \end{matrix} \]

Center of Mass \( = (\bar{x}, \bar{y}) = \left( \frac{1}{M} M_y, \frac{1}{M} M_x \right) \)

Page 4

Example 1

\(D\) is the region below with constant/uniform Density.

A 2D Cartesian coordinate graph illustrating a rectangular region D. The horizontal axis represents x and the vertical axis represents y. The region D is a shaded gray rectangle defined by the bounds x from 1 to 2 and y from 0 to pi. The four vertices of the rectangle are explicitly labeled with coordinates: bottom-left at (1, 0), bottom-right at (2, 0), top-left at (1, pi), and top-right at (2, pi). A solid red circle is located in the geometric center of the rectangle, marking the predicted center of mass.
Visual Description: A 2D Cartesian coordinate graph illustrating a rectangular region D. The horizontal axis represents x and the vertical axis represents y. The region D is a shaded gray rectangle defined by the bounds x from 1 to 2 and y from 0 to pi. The four vertices of the rectangle are explicitly labeled with coordinates: bottom-left at (1, 0), bottom-right at (2, 0), top-left at (1, pi), and top-right at (2, pi). A solid red circle is located in the geometric center of the rectangle, marking the predicted center of mass.

Guess: \(C.O.M = \left( \frac{1}{2}, \frac{\pi}{2} \right)\)

Let \(\rho(x, y) = k\)

\[ M = \iint_D \rho(x, y) \, dA = \int_0^\pi \int_1^2 k \, dx \, dy = k\pi \] \[ M_y = \iint_D x \cdot \rho(x, y) \, dA = \int_0^\pi \int_1^2 kx \, dx \, dy = \frac{3k\pi}{2} \] \[ M_x = \iint_D y \cdot \rho(x, y) \, dA = \int_0^\pi \int_1^2 ky \, dx \, dy = k \times 1 \times \frac{\pi^2}{2} = \frac{k\pi^2}{2} \] \[ C.O.M = \left( \frac{1}{M} M_y, \frac{1}{M} M_x \right) = \left( \frac{1}{k\pi} \frac{3k\pi}{2}, \frac{1}{k\pi} \frac{k\pi^2}{2} \right) = \left( \frac{3}{2}, \frac{\pi}{2} \right) \]

Page 5

#### Example 2
\(D\) is the region below with Density \(\rho(x, y) = x \sin y\)

Guess: C.O.M tilted towards Right

A 2D plot on a Cartesian coordinate system illustrating the region D. The region is a shaded grey rectangle bounded by the lines x = 1, x = 2, y = 0, and y = pi. The four vertices are labeled with their coordinates: (1, 0), (2, 0), (2, pi), and (1, pi). A single red dot is plotted within the shaded region, positioned slightly to the right of the horizontal center and exactly at the vertical center, representing the Center of Mass (C.O.M.). The horizontal axis represents x and the vertical axis represents y.
Visual Description: A 2D plot on a Cartesian coordinate system illustrating the region D. The region is a shaded grey rectangle bounded by the lines x = 1, x = 2, y = 0, and y = pi. The four vertices are labeled with their coordinates: (1, 0), (2, 0), (2, pi), and (1, pi). A single red dot is plotted within the shaded region, positioned slightly to the right of the horizontal center and exactly at the vertical center, representing the Center of Mass (C.O.M.). The horizontal axis represents x and the vertical axis represents y.
\[ M = \iint_D \rho(x,y) \, dA = \int_0^\pi \int_1^2 x \sin y \, dx \, dy = 3. \] \[ M_y = \iint_D x \rho(x,y) \, dA = \int_0^\pi \int_1^2 x^2 \sin y \, dx \, dy = \frac{14}{3} \] \[ M_x = \iint_D y \rho(x,y) \, dA = \int_0^\pi \int_1^2 x \cdot y \sin y \, dx \, dy = \frac{3}{2} \int_0^\pi y \sin y \, dy \]

\(\downarrow\) integrate by parts

\[ = \frac{3\pi}{2} \] \[ \text{C.O.M} = \left( \frac{1}{3} \cdot \frac{14}{3}, \frac{1}{3} \cdot \frac{3\pi}{2} \right) = \left( \frac{14}{9}, \frac{\pi}{2} \right) \approx \left( 1.6, \frac{\pi}{2} \right) \]

Page 6

Aside: integration by parts

\[ \int u \, dv = uv - \int v \, du \]
Example calculation: \[ \int y \sin y \, dy \quad \rightsquigarrow \quad u = y, \, du = dy \] \[ \phantom{\int y \sin y \, dy} \quad dv = \sin y \, dy \rightsquigarrow v = -\cos y \] Applying the formula to the definite integral: \[ \int_{0}^{\pi} y \sin y \, dy = -y \cos y - \int (-\cos y) \, dy = \left. -y \cos y + \sin y \right|_{0}^{\pi} \] \[ = (-\pi \cos \pi + \sin \pi) - (0 + 0) \] \[ = \pi \]

Page 7

Example 3

\(D\) is the region below with density proportional to distance from the \(y\)-axis.

A 2D coordinate graph illustrating the region D in the first quadrant of a Cartesian plane. The region is bounded above by the line y = x and below by the parabolic curve y = x^2. These two curves intersect at the origin (0,0) and at the point (1,1). The area between them is shaded grey to represent the region D. A red point is placed within the shaded area, and through it pass a horizontal green line and a vertical blue line. Annotations include a horizontal arrow labeled 'x' indicating the distance from the y-axis to the point, and a vertical arrow labeled 'y' indicating the distance from the x-axis. The axes and curves are clearly labeled.
Visual Description: A 2D coordinate graph illustrating the region D in the first quadrant of a Cartesian plane. The region is bounded above by the line y = x and below by the parabolic curve y = x^2. These two curves intersect at the origin (0,0) and at the point (1,1). The area between them is shaded grey to represent the region D. A red point is placed within the shaded area, and through it pass a horizontal green line and a vertical blue line. Annotations include a horizontal arrow labeled 'x' indicating the distance from the y-axis to the point, and a vertical arrow labeled 'y' indicating the distance from the x-axis. The axes and curves are clearly labeled.

The density function is given by: \[ \rho(x, y) = kx \]

The total mass \(M\) of the region is calculated as: \[ M = \iint_D \rho \, dA = \int_0^1 \int_{x^2}^x k \cdot x \, dy \, dx \] \[ = k \cdot \int_0^1 x(x - x^2) \, dx = k \left[ \frac{1}{3} - \frac{1}{4} \right] = k/12 \]

The moments about the \(y\)-axis and \(x\)-axis are calculated as follows: \[ M_y = \int_0^1 \int_{x^2}^x x \cdot kx \, dy \, dx = k \cdot \int_0^1 x^2(x - x^2) \, dx = k \left( \frac{1}{4} - \frac{1}{5} \right) = k/20 \] \[ M_x = \int_0^1 \int_{x^2}^x y \cdot kx \, dy \, dx = \frac{k}{2} \int_0^1 x(x^2 - x^4) \, dx = \frac{k}{2} \left( \frac{1}{4} - \frac{1}{6} \right) = k/24 \]

The center of mass (C.O.M) is then: \[ \text{C.O.M} = \left( \frac{k/20}{k/12}, \frac{k/24}{k/12} \right) = (0.6, 0.5) \]

Page 8

Mass, Moment & Center of Mass for 3D objects

A diagram showing a 3D Cartesian coordinate system with axes labeled x, y, and z. An irregularly shaped region, labeled D, is situated in the 3D space. A small rectangular prism, shaded in red, represents a volume element within D. Horizontal dashed lines divide the region D into thin horizontal slices, illustrating the conceptual setup for a triple integral over the volume.
Visual Description: A diagram showing a 3D Cartesian coordinate system with axes labeled x, y, and z. An irregularly shaped region, labeled D, is situated in the 3D space. A small rectangular prism, shaded in red, represents a volume element within D. Horizontal dashed lines divide the region D into thin horizontal slices, illustrating the conceptual setup for a triple integral over the volume.

density \(= \rho(x, y, z)\)

Mass of

A small red sketch of a 3D rectangular prism, representing a differential volume element \(\Delta V\).
Visual Description: A small red sketch of a 3D rectangular prism, representing a differential volume element \(\Delta V\).
\(= \rho(x, y, z) \Delta V\)

Mass of \(D = \iiint_D \rho(x, y, z) \, dV\)

\(M_{xy}\) of;

A small red sketch of a 3D rectangular prism, representing a differential volume element.
Visual Description: A small red sketch of a 3D rectangular prism, representing a differential volume element.
\(=\) Moment about xy plane \(= z \cdot \rho(x, y, z) \Delta V\)

\(M_{xy} \text{ of } D = \iiint_D z \rho(x, y, z) \, dV\)

\(M_{yz} \text{ of } D = \iiint_D x \rho(x, y, z) \, dV\)

\(M_{xz} \text{ of } D = \iiint_D y \rho(x, y, z) \, dV\)

C.O.M. \(= \left( \frac{1}{M} M_{yz}, \frac{1}{M} M_{xz}, \frac{1}{M} M_{xy} \right)\)

Page 9

Second Moment / Moment of inertia about an axis

Moment = distance \(\times\) Mass

Second Moment = \((\text{distance})^2 \times \text{Mass}\)

\[ I_{\text{axis}} = \iiint\limits_D d^2 \rho(x, y, z) \, dV \]

Where:

  • \(\rho(x, y, z)\) is the density.
  • \(d\) is the distance of a point in \(D\) to the given axis.
A 3D Cartesian coordinate system illustrating the distance from a point to an axis. The z-axis is a vertical blue line. A point in space is labeled (x, y, z). A perpendicular line segment labeled 'd' connects the point (x, y, z) to the z-axis. The point of intersection on the z-axis is labeled (0, 0, z). The x and y axes are shown extending from the origin at the base of the z-axis, providing the 3D context.
Visual Description: A 3D Cartesian coordinate system illustrating the distance from a point to an axis. The z-axis is a vertical blue line. A point in space is labeled (x, y, z). A perpendicular line segment labeled 'd' connects the point (x, y, z) to the z-axis. The point of intersection on the z-axis is labeled (0, 0, z). The x and y axes are shown extending from the origin at the base of the z-axis, providing the 3D context.

distance of \((x, y, z)\) to z-axis = \(\sqrt{x^2 + y^2}\)

\[ I_z = \iiint\limits_D (x^2 + y^2) \rho(x, y, z) \, dV \]