8.4 Varying Densitymathsci2.appstate.edu/~sjg/class/1120/1120density84.pdfSubstance mass with...

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8.4 Varying Density Density over length, area or volume, changing only in 1 dimension (Calc III for others) Dr. Sarah Math 1120: Calculus and Analytic Geometry II

Transcript of 8.4 Varying Densitymathsci2.appstate.edu/~sjg/class/1120/1120density84.pdfSubstance mass with...

Page 1: 8.4 Varying Densitymathsci2.appstate.edu/~sjg/class/1120/1120density84.pdfSubstance mass with density varying over a volume g/cm3 Population quantity like people/square mile, bacteria/cc.

8.4 Varying DensityDensity over length, area or volume, changing only in 1dimension (Calc III for others)

Substance mass with density varying over a volume g/cm3

Population quantity like people/square mile, bacteria/cc.Slice so density is approximately constant:

If δ = f (x), then slice ⊥ x , for constant density slicesmass=

∑density · length (∆x), area, or volume

Key: figure out slicing variable, then∫δ· length or

∫δ·area

or∫δ·volume in that variable

Dr. Sarah Math 1120: Calculus and Analytic Geometry II

Page 2: 8.4 Varying Densitymathsci2.appstate.edu/~sjg/class/1120/1120density84.pdfSubstance mass with density varying over a volume g/cm3 Population quantity like people/square mile, bacteria/cc.

8.4 Varying DensityDensity over length, area or volume, changing only in 1dimension (Calc III for others)Substance mass with density varying over a volume g/cm3

Population quantity like people/square mile, bacteria/cc.

Slice so density is approximately constant:If δ = f (x), then slice ⊥ x , for constant density slices

mass=∑

density · length (∆x), area, or volumeKey: figure out slicing variable, then

∫δ· length or

∫δ·area

or∫δ·volume in that variable

Dr. Sarah Math 1120: Calculus and Analytic Geometry II

Page 3: 8.4 Varying Densitymathsci2.appstate.edu/~sjg/class/1120/1120density84.pdfSubstance mass with density varying over a volume g/cm3 Population quantity like people/square mile, bacteria/cc.

8.4 Varying DensityDensity over length, area or volume, changing only in 1dimension (Calc III for others)Substance mass with density varying over a volume g/cm3

Population quantity like people/square mile, bacteria/cc.Slice so density is approximately constant:

If δ = f (x), then slice ⊥ x , for constant density slicesmass=

∑density · length (∆x), area, or volume

Key: figure out slicing variable, then∫δ· length or

∫δ·area

or∫δ·volume in that variable

Dr. Sarah Math 1120: Calculus and Analytic Geometry II

Page 4: 8.4 Varying Densitymathsci2.appstate.edu/~sjg/class/1120/1120density84.pdfSubstance mass with density varying over a volume g/cm3 Population quantity like people/square mile, bacteria/cc.

8.4 Varying DensityDensity over length, area or volume, changing only in 1dimension (Calc III for others)Substance mass with density varying over a volume g/cm3

Population quantity like people/square mile, bacteria/cc.Slice so density is approximately constant:

If δ = f (x), then slice ⊥ x , for constant density slicesmass=

∑density · length (∆x), area, or volume

Key: figure out slicing variable, then∫δ· length or

∫δ·area

or∫δ·volume in that variable

Dr. Sarah Math 1120: Calculus and Analytic Geometry II

Page 5: 8.4 Varying Densitymathsci2.appstate.edu/~sjg/class/1120/1120density84.pdfSubstance mass with density varying over a volume g/cm3 Population quantity like people/square mile, bacteria/cc.

8.4 Varying DensityDensity over length, area or volume, changing only in 1dimension (Calc III for others)Substance mass with density varying over a volume g/cm3

Population quantity like people/square mile, bacteria/cc.Slice so density is approximately constant:

If δ = f (x), then slice ⊥ x , for constant density slicesmass=

∑density · length (∆x), area, or volume

Key: figure out slicing variable, then∫δ· length or

∫δ·area

or∫δ·volume in that variable

Dr. Sarah Math 1120: Calculus and Analytic Geometry II

Page 6: 8.4 Varying Densitymathsci2.appstate.edu/~sjg/class/1120/1120density84.pdfSubstance mass with density varying over a volume g/cm3 Population quantity like people/square mile, bacteria/cc.

Clicker Question

1. If δ = f (h) kg/m3, where h is the distance of a slice of acylindrical tank of radius 5m and length 21m buried 3 m belowground, then the total mass is:

a)∫ 13

3 2πhδ(h) dh

b)∫ 13

3 πh2δ(h) dh

c)∫ 13

3 δ(h)21× 2√

52 − (8− h)2 × dhd) other

∫ 133 δ(h) volume of a slice =

∫ 133 δ(h)21× 2

√52 − (8− h)2 × dh

Dr. Sarah Math 1120: Calculus and Analytic Geometry II

Page 7: 8.4 Varying Densitymathsci2.appstate.edu/~sjg/class/1120/1120density84.pdfSubstance mass with density varying over a volume g/cm3 Population quantity like people/square mile, bacteria/cc.

Clicker Question

1. If δ = f (h) kg/m3, where h is the distance of a slice of acylindrical tank of radius 5m and length 21m buried 3 m belowground, then the total mass is:

a)∫ 13

3 2πhδ(h) dh

b)∫ 13

3 πh2δ(h) dh

c)∫ 13

3 δ(h)21× 2√

52 − (8− h)2 × dhd) other∫ 13

3 δ(h) volume of a slice =∫ 13

3 δ(h)21× 2√

52 − (8− h)2 × dh

Dr. Sarah Math 1120: Calculus and Analytic Geometry II

Page 8: 8.4 Varying Densitymathsci2.appstate.edu/~sjg/class/1120/1120density84.pdfSubstance mass with density varying over a volume g/cm3 Population quantity like people/square mile, bacteria/cc.

Dr. Sarah Math 1120: Calculus and Analytic Geometry II

Page 9: 8.4 Varying Densitymathsci2.appstate.edu/~sjg/class/1120/1120density84.pdfSubstance mass with density varying over a volume g/cm3 Population quantity like people/square mile, bacteria/cc.

History and Applications

Archimedes and King’s gold1798 Henry Cavendish density ‘weighing the world’Sir Isaac Newton: aerodynamic drag is proportional to airdensity, cross sectional area and v2

protein in kidneypopulation density to model infectious diseasedensity of universe and geometryProbability Density Function (PDF)Cumulative Distribution Function (CDF)

Dr. Sarah Math 1120: Calculus and Analytic Geometry II