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Wize University Calculus 1 Textbook > Applications of Integration for Physical Science

Mass and Density

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Mass and Density

The Mass of an object measures the amount of material of the object. The Density of an object measures the mass per volume of the object.

Density

For an object with mass MMand volume VV, having constant density ρ\rho

ρ=MV\boxed{\displaystyle \rho=\frac{M}{V}}

Mass Integrals

For an object with cross-sectional area A(x)A(x)and density ρ(x)\rho(x)on the interval [a,b][a,b], the mass MMis
M=∫abρ(x)A(x)dx\boxed{\displaystyle M=\int_a^b\rho(x)A(x)dx}

Wize Concept
For 3-D solids, A(x)A(x) can be determined using the methods of volume of revolution by cross sections.

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Example: Mass and Density

The region (shown below) is defined by y=x2y=x^2 and y=−x2+2y=-x^2+2 and has density ρ(x)=x+1\rho(x)=x+1. Find the Mass of the shaded region.



M=∫abρ(x)A(x)dx\boxed{\displaystyle M=\int_a^b\rho(x)A(x)dx}

To determine our limits of integration, we must find the intersection points of the curves.

x2=−x2+2  ⟺  2x2=2  ⟺  x2=1  ⟺  x=±1x^2=-x^2+2 \iff 2x^2=2 \iff x^2 = 1 \iff x=\pm1

ρ(x)=x+1\rho(x)=x+1 and the cross sectional areaA(x)=Top Curve−Bottom Curve=(−x2+2)−(x2)A(x)= \text{Top Curve}- \text{Bottom Curve}=(-x^2+2)-(x^2)

A(x)=−2x2+2A(x)=-2x^2+2

M=∫−11(x+1)(−2x2+2)dx\displaystyle M=\int_{-1}^1(x+1)(-2x^2+2)dx

=∫−11(−2x3−2x2+2x+2)dx\displaystyle =\int_{-1}^1(-2x^3-2x^2+2x+2)dx

=(−2x44−2x33+2x22+2x)∣−11\displaystyle =\bigg(\frac{-2x^4}{4}-\frac{2x^3}{3}+\frac{2x^2}{2}+2x \bigg) \bigg |_{-1}^1

=(−2(1)44−2(1)33+2(1)22+2(1))−(−2(−1)44−2(−1)33+2(−1)22+2(−1))\displaystyle =\bigg(\frac{-2(1)^4}{4}-\frac{2(1)^3}{3}+\frac{2(1)^2}{2}+2(1) \bigg) -\displaystyle \bigg(\frac{-2(-1)^4}{4}-\frac{2(-1)^3}{3}+\frac{2(-1)^2}{2}+2(-1) \bigg)

=4−43=83\displaystyle =4-\frac{4}{3}= \boxed{\frac{8}{3}}
Snow and ice is piled into a cone which is 5 m high and has a circular base of radius 10 m. The snow-and-ice mixture near the base has been compressed over time and so the pile is denser near the bottom. To be precise, at a height of yy meters above ground, the density is
ρ(y)=250−50 ykg/m3\rho(y)=250-50~y\quad kg/m^3
Find the total mass (in kilograms) of the cone.


Extra Practice
Density of Liquids
An unknown liquid compound is formed in a research lab. The researcher measures that a flask containing 13.6 mL of the liquid weighs 426.2 g and the empty flask weighs 408.8 g. What is the density of the unknown liquid?
Applications of Integration: Mass and Density
Snow and ice is piled into a cone which is 5 m high and has a circular base of radius 10 m. The snow-and-ice mixture near the base has been compressed over time and so the pile is denser near the bottom. To be precise, at a height of yy meters above ground, the density is
ρ(y)=250−50 ykg/m3\rho(y)=250-50~y\quad kg/m^3
Find the total mass (in kilograms) of the cone.
Density of Liquids
An unknown liquid compound is formed in a research lab. The researcher measures that a flask containing 13.6 mL of the liquid weighs 426.2 g and the empty flask weighs 408.8 g. What is the density of the unknown liquid?