Appendix E Examples
1.1 Antiderivatives
Example 1.1.3 Basic antiderivative computations
Example 1.1.4 Antiderivatives depend on intervals
Example 1.1.6 Less elementary antiderivative computations
Example 1.1.9 Leaking water tank
Example 1.1.10 Initial value problem
Example 1.1.13 Indefinite integral formulas
1.2 Estimating area
Example 1.2.3 Estimating area: \(f(x)=1-x^3\)
Example 1.2.4 Estimating area: \(f(x)=1-\abs{x}^3\)
Example 1.2.7 Estimating displacement
1.3 Riemann sums
Example 1.3.2 Sequence of primes
Example 1.3.3 Sinusoidal series
Example 1.3.6 Closed form of summation
1.4 Definite integral
Example 1.4.3 Integral of linear function
Example 1.4.4 Integral of cubic
Example 1.4.7 Integral of cubic (cont.)
Example 1.4.11 Integral of linear function (cont.)
1.5 Fundamental theorem of calculus
Example 1.5.6 Computing integrals with antiderivatives
Example 1.5.7 Signed area using FTC II
Example 1.5.9 Leaking water tank revisited
1.6 Fundamental theorem (cont.)
Example 1.6.3 Integral functions
Example 1.6.6 FTC I+chain rule
1.7 Substitution
Example 1.7.2 Substitution:straightforward
Example 1.7.5 Substitution: less straightforward
Example 1.7.7 Definite integral substitution: 2-step technique
1.8 More substitution; area between curves
Example 1.8.2 Definite integral substitution: streamlined
Example 1.8.6 Area between parabola and line
Example 1.8.8 Area between parabolas
1.9 Volume via cross sections
Example 1.9.3 Volume of sphere
Example 1.9.4 Volume of cone
Example 1.9.7 Disk example
Example 1.9.8 Washer example
1.10 Inverse functions
Example 1.10.6 Computing inverse
Example 1.10.10 Derivative of inverse
1.12 Exponential functions
Example 1.12.9 Solving exponential equations
Example 1.12.10 Exponential and logarithmic derivatives
Example 1.12.11 Exponential and logarithmic integrals
1.13 Separable differential equations
Example 1.13.2 Exponential growth/decay
Example 1.13.6 Separation of variables
Example 1.13.7 Newton’s law of cooling
1.14 L’Hôpital’s rule
Example 1.14.2 Indeterminate forms
Example 1.14.6 L’Hôpital’s rule
Example 1.14.8 Indeterminate form limit
1.15 More on indeterminate forms
Example 1.15.3 More indeterminate forms
1.16 Inverse trigonometric functions
Example 1.16.4 Computing with inverse trig functions
Example 1.16.6 Solving trig equations
Example 1.16.8 Derivatives of inverse trig functions
Example 1.16.9 Limit computation
Example 1.16.10 Inverse trig functions as antiderivatives
Example 1.16.11 Inverse trig functions as antiderivatives
1.17 Integration strategies
Example 1.17.2 Vertex form
Example 1.17.3 Polynomial division
Example 1.17.4 Exponential substitution
Example 1.17.5 Exponential substitution (again)
Example 1.17.6 Trig identity
1.18 Integration by parts
Example 1.18.3 Classic by parts
Example 1.18.5 Iterated by parts
Example 1.18.6 Surprising by parts
Example 1.18.7 Rational function
Example 1.18.8 Inverse trig
Example 1.18.9 By parts and algebra
1.19 Trigonometric integrals
Example 1.19.4 Odd sine power
Example 1.19.5 Even powers
Example 1.19.6 Even secant power
Example 1.19.7 Odd tangent power
Example 1.19.8 Even tangent power, odd secant power
Example 1.19.9 Tangent power
1.20 Trigonometric substitution
Example 1.20.8 Sine substitution
Example 1.20.9 Area of circle
Example 1.20.10 Tangent substitution
Example 1.20.11 Secant substitution
Example 1.20.12 Secant substitution: definite
Example 1.20.13 Secant substitution: indefinite
1.21 Rational functions
Example 1.21.6 Rational function
Example 1.21.7 Long division first
Example 1.21.8 Three distinct roots
Example 1.21.9 Two irreducible quadratics
Example 1.21.10 With substitution
1.22 Numerical integration: techniques
Example 1.22.4 Estimating \(\ln 4\)
Example 1.22.5 Estimating \(\pi\)
1.23 Numerical integration: error bounds
Example 1.23.2 Estimating \(\ln 4\text{:}\) error bounds
Example 1.23.3 Estimating \(\pi\text{:}\) error bounds
1.24 Improper integrals
Example 1.24.3 Type I: half-infinite
Example 1.24.4 Type I: \(p\)-test
Example 1.24.5 Type I: real line integral
Example 1.24.9 Improper: type II
Example 1.24.10 Improper: type II
Example 1.24.11 Improper: type II
1.25 Improper integrals: convergence tests
Example 1.25.2 Direct comparison test
Example 1.25.3 Direct comparison test
Example 1.25.7 Limit comparison test
Example 1.25.8 Limit comparison test
Example 1.25.9 Limit comparison test
1.26 First-order linear differential equations
Example 1.26.4 Exponential change revisited
Example 1.26.5 Non-separable example
Example 1.26.6 Initial value
1.27 Modeling with differential equations
Example 1.27.2 Mixing problem
Example 1.27.3 Trout population
Example 1.27.4 Spreading rumor
