2.7 Natural log as area · Logarithmic, Exponential, and Hyperbolic Models · Visual Calculus

2.7ln as area

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Chapter 2 · Applications of Integration

2.1-2.2Areas & Slicing

2.3-2.4Shells & Arc Length

2.5Physical Applications

2.6Centers of Mass

2.7-2.9Log, Exp & Hyperbolic

2.7ln as area 2.7Exp substitution 2.8Exponential growth 2.8Newton cooling 2.9Hyperbolic definitions 2.9Circle vs hyperbola 2.9Hyperbolic identity 2.9Hyperbolic graphs 2.9Hyperbolic derivatives 2.9Hyperbolic integrals 2.9Inverse graphs 2.9Inverse derivatives 2.9Inverse integrals 2.9Catenary shape 2.9Cable length

Chapter 3 · Techniques of Integration

3.1-3.4Techniques I

3.5-3.7Techniques II

Chapter 5 · Sequences and Series

5.1-5.3Sequences & Series I

5.4-5.6Sequences & Series II

Chapter 7 · Parametric Equations and Polar Coordinates

7.1-7.2Parametric Curves

7.3-7.4Polar Curves

Chapter 6 · Power Series

6.1-6.2Power Series I

6.3-6.4Power Series II

Home Calculus IISections 2.7-2.92.7 Natural log as area

2.7

Natural log as area

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SymPy check = 1.0986122887

SymPy checks

Natural log area1.0986122887

Exponential substitution area0.6321205588

Growth threshold time310.7304049211

Coffee serving start2.5569455632

Integral of cosh over [-1, 1]2.3504023873

Unit sector area0.5

Hyperbolic integral1.1752011936

Integral of tanh over [0, 1]0.4337808305

Integral of sech^2 over [0, 1]0.761594156

Integral of sinh over [0, 1]0.5430806348

Integral of 1/(1-x^2) over [0, 1/2]0.5493061443

Integral of 1/sqrt(1+x^2) over [0, 1]0.881373587

Integral of 1/sqrt(4x^2-1) over [1/2, 1]0.6584789485

Arc length of 2 cosh(x/2) over [-2, 2]4.7008047746

Catenary cable length42.5855891019