Math 146 - Calc II
Review for Test 1
Tests will consist in 5 problems. You are allowed one page with formulas, but with no examples or
solutions. Please review the theory first, and work the suggested problems. Then continue with the other
homework problems, and review the proofs of the theorems.
Ch. 6 Integrals
6.1 Areas between two curves
6.2 Volumes
- Integral of cross-section area
- Solids of revolution (special description of solid => special cross-section):
1) Disk method (cross-sections = disks)
2) Washer method (c-s = washers)
6.3 Volumes by cylindrical shells
- Like “Volume by slicing”, except using cylindrical shells
- Integral of cross-section area (cylinders)
6.4 Work
- Work done by a constant force: W=F x d, work done by a variable force in 1-dimension: dW=F dx
- Elastic force & Hooks law for springs (F=-kx)
- Gravity: pulling ropes & pumping liquids from containers (F=mg) dW=dG x d
6.5 Average value of a Function
- Definition, Mean Value Theorem for Integrals: f_ave=f(c)
Ch.7 Techniques of integration
- Integration by parts: formula, twice integration by parts and equation for I, recursive formulas.
- Trigonometric integrals: sin and cos w. even/odd powers, tan and sec, sin(nx)cos(mx)
- Trigonometric substitution for square roots of quadratic functions
- Integration by Partial Fractions Decomposition: proper fractions, linear / quadratic factors with distinct
or repeated factors
- Strategy of integration: simplify, u-substitutions or change of variables, classify and apply a
corresponding method, tables of integrals and CAS.
Math 146 - Calc II
Review for Test 2
Lengths of plane curves: Differential formula: L= ds, ds=(x’2+y’2)½ dt or ds=(1+y’2)½ dt
Areas of surfaces of revolution: Surface area:  L(x)ds, L(x)=2 r(x)
Applications to physics and Engineering:
- Hydrostatic force and pressure,
- Center of Mass, Torque of a mass: mgx; moment: mx
- Wires and Thin rods; thin flat plate (1-integrals)
Applications to economics and biology: consumer surplus, cardiac output.
Probability: density function, random variable, P(a<X<b), mean value, normal distribution.
9.1 Modeling with DE
- Population models: natural law of growth
- General 1st ODE, solution, initial condition, initial value problem
9.2 Direction fields and Euler’s method
- Slope fields: viewing solution curves – use calculator
- Euler’ method
9.3 Separable DE:
- Def., solution, equilibrium points/constant solutions
- Applications: orthogonal trajectories, mixing problems
9.4 Models of population growth:
- Natural growth, logistic model, harvesting model: def., properties.
9.5 First-order linear DE
- Standard form, integrating factor, solving linear equations
- RL Circuits
9.6 Predator-Prey systems
- Def. , equilibrium solutions, phase portrait, elimination of time
Math 146 - Calc II
Review for Test 3
Ch. 10 Parametric equations and polar coordinates
- Parametric curves, tangent line, arc length, areas
- Polar coordinates, polar curves, tangents to polar curves, areas and lengths
- Conic sections: definition, equations, polar coordinates, Kepler’s laws
Ch. 11 Infinite sequences and series
11.1 Sequences: Infinite sequences, convergence and divergence, diverges to infinity, calculating limits
of sequences, monotonic sequence theorem
11.2 Infinite series:
- Geometric series, convergent/divergent series, nth-term test for divergence
- Combining series, adding or deleting terms, reindexing
11.3 The integral test: non-decreasing partial sums, the integral test, remainder estimate
11.4 Comparison tests: the comparison test, limit comparison test
11.5, 11.6 Alternating series, absolute and conditional convergence
- Alternating series, harmonic series, The Alternating Series Estimation Theorem
- Absolute and conditional convergence, rearranging terms
11.7 Strategy for testing series
11.8, 11.9 Power series:
- Power series and convergence, radius of convergence of a power series, representations of functions
- Term-by-term differentiation and integration
11.10 Taylor and MacLaurin series:
- Series representations, Taylor and MacLaurin Series, Taylor polynomials
- Multiplication and division of power series
11.11 Applications of Taylor polynomials: estimating the remainder