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Essential Concepts

Physical Applications

• Several physical applications of the definite integral are common in engineering and physics.
• Definite integrals can be used to determine the mass of an object if its density function is known.
• Work can also be calculated from integrating a force function, or when counteracting the force of gravity, as in a pumping problem.
• Definite integrals can also be used to calculate the force exerted on an object submerged in a liquid.

Moments and Centers of Mass

• Mathematically, the center of mass of a system is the point at which the total mass of the system could be concentrated without changing the moment. Loosely speaking, the center of mass can be thought of as the balancing point of the system.
• For point masses distributed along a number line, the moment of the system with respect to the origin is [latex]M=\displaystyle\sum_{i=1}^{n} {m}_{i}{x}_{i}.[/latex] For point masses distributed in a plane, the moments of the system with respect to the [latex]x[/latex]– and [latex]y[/latex]-axes, respectively, are [latex]{M}_{x}=\displaystyle\sum_{i=1}{n} {m}_{i}{y}_{i}[/latex] and [latex]{M}_{y}=\displaystyle\sum_{i=1}{n} {m}_{i}{x}_{i},[/latex] respectively.
• For a lamina bounded above by a function [latex]f(x),[/latex] the moments of the system with respect to the [latex]x[/latex]– and [latex]y[/latex]-axes, respectively, are [latex]{M}_{x}=\rho {\displaystyle\int }_{a}^{b}\frac{{\left[f(x)\right]}^{2}}{2}dx[/latex] and [latex]{M}_{y}=\rho {\displaystyle\int }_{a}^{b}xf(x)dx.[/latex]
• The [latex]x[/latex]– and [latex]y[/latex]-coordinates of the center of mass can be found by dividing the moments around the [latex]y[/latex]-axis and around the [latex]x[/latex]-axis, respectively, by the total mass. The symmetry principle says that if a region is symmetric with respect to a line, then the centroid of the region lies on the line.
• The theorem of Pappus for volume says that if a region is revolved around an external axis, the volume of the resulting solid is equal to the area of the region multiplied by the distance traveled by the centroid of the region.

Key Equations

• Mass of a one-dimensional object
  [latex]m={\displaystyle\int }_{a}^{b}\rho (x)dx[/latex]
• Mass of a circular object
  [latex]m={\displaystyle\int }_{0}^{r}2\pi x\rho (x)dx[/latex]
• Work done on an object
  [latex]W={\displaystyle\int }_{a}^{b}F(x)dx[/latex]
• Hydrostatic force on a plate
  [latex]F={\displaystyle\int }_{a}^{b}\rho w(x)s(x)dx[/latex]
• Mass of a lamina
  [latex]m=\rho {\displaystyle\int }_{a}^{b}f(x)dx[/latex]
• Moments of a lamina
  [latex]{M}_{x}=\rho {\displaystyle\int }_{a}^{b}\frac{{\left[f(x)\right]}^{2}}{2}dx\text{ and }{M}_{y}=\rho {\displaystyle\int }_{a}^{b}xf(x)dx[/latex]
• Center of mass of a lamina
  [latex]\overline{x}=\frac{{M}_{y}}{m}\text{ and }\overline{y}=\frac{{M}_{x}}{m}[/latex]

Glossary

center of mass

the point at which the total mass of the system could be concentrated without changing the moment

centroid

the centroid of a region is the geometric center of the region; laminas are often represented by regions in the plane; if the lamina has a constant density, the center of mass of the lamina depends only on the shape of the corresponding planar region; in this case, the center of mass of the lamina corresponds to the centroid of the representative region

density function

a density function describes how mass is distributed throughout an object; it can be a linear density, expressed in terms of mass per unit length; an area density, expressed in terms of mass per unit area; or a volume density, expressed in terms of mass per unit volume; weight-density is also used to describe weight (rather than mass) per unit volume

Hooke’s law

this law states that the force required to compress (or elongate) a spring is proportional to the distance the spring has been compressed (or stretched) from equilibrium; in other words, [latex]F=kx,[/latex] where [latex]k[/latex] is a constant

hydrostatic pressure

the pressure exerted by water on a submerged object

lamina

a thin sheet of material; laminas are thin enough that, for mathematical purposes, they can be treated as if they are two-dimensional

moment

if [latex]n[/latex] masses are arranged on a number line, the moment of the system with respect to the origin is given by [latex]M=\displaystyle\sum_{i=1}{n} {m}_{i}{x}_{i};[/latex] if, instead, we consider a region in the plane, bounded above by a function [latex]f(x)[/latex] over an interval [latex]\left[a,b\right],[/latex] then the moments of the region with respect to the [latex]x[/latex]– and [latex]y[/latex]-axes are given by [latex]{M}_{x}=\rho {\displaystyle\int }_{a}^{b}\frac{{\left[f(x)\right]}^{2}}{2}dx[/latex] and [latex]{M}_{y}=\rho {\displaystyle\int }_{a}^{b}xf(x)dx,[/latex] respectively

symmetry principle

the symmetry principle states that if a region R is symmetric about a line [latex]l[/latex], then the centroid of R lies on [latex]l[/latex]

theorem of Pappus for volume

this theorem states that the volume of a solid of revolution formed by revolving a region around an external axis is equal to the area of the region multiplied by the distance traveled by the centroid of the region

work

the amount of energy it takes to move an object; in physics, when a force is constant, work is expressed as the product of force and distance

Study Tips

Mass and Density

• Visualize the partitioning process for both rod and disk
• Remember the difference between linear and area density
• Connect these formulas to earlier integration techniques (e.g., shells method)
• Consider how these formulas might extend to three-dimensional objects

Work Done by a Force

• Visualize the force as the height of a curve and work as the area under this curve
• Remember the sign convention: positive work when force is in the direction of motion
• For spring problems, identify the equilibrium position and direction of stretch/compression
• Always check units: Work should be in Joules (N·m)

Work Done in Pumping

• Practice with different tank shapes (cylinder, cone, etc.)
• Remember to account for partial filling/emptying
• Be consistent with units (metric or imperial)
• Visualize the process of lifting water layer by layer
• Review volume formulas for various geometric shapes

Hydrostatic Force and Pressure

• Practice setting up integrals for various shapes (rectangles, triangles, etc.)
• Visualize the pressure increasing with depth
• Consider how the width of the object changes with depth

Center of Mass and Moments

• Visualize the balance point for simple systems
• Practice with both one and two-dimensional problems
• Remember the difference between [latex]x[/latex] and [latex]y[/latex] moments in 2D
• Check your answer by considering symmetry and intuition

Center of Mass of Thin Plates

• Visualize the lamina and its bounding curves
• Remember that density cancels out in final calculations
• Connect these concepts to earlier work on centers of mass for point systems

Center of Mass of a Region Bounded by Two Functions

• Visualize the region between the two functions
• Practice finding intersection points of functions to determine integration limits
• Review integration techniques, especially for more complex functions

The Symmetry Principle

• Practice identifying lines of symmetry in various shapes
• Visualize the revolution of planar regions around an axis
• Compare results from Pappus’s Theorem with other volume calculation methods
• Remember that symmetry can simplify both centroid and volume calculations