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Integration
- Introduce the Fundamental Theorem of Calculus.
- Use the Fundamental Theorem to find the area under a closed region.
- Be able to change the limits of definite integrals in a U-Substitution.
- State the Mean Value Theorem for Integrals.
- Define the average value of a function.
- Integrate even and odd functions.
- Introduce the logarithmic integral.
- Develop and use the six basic trigonometric integrals.
- Develop and use the integral rules for exponential functions.
- Find the area bounded by exponential functions.
- Be able to integrate exponential functions to any base.
- Use the law of exponential growth and decay.
- Develop and use the integration rules of inverse trigonometric functions.
- Be able to integrate improper rational functions.
- Develop "completing the square" as a technique of integration.
Applications of Integration
- Find the area of a region between two curves.
- Introduce horizontal representative rectangles to find area.
- Determine the volume of a solid of revolution using the Disc Method.
- Use subtraction formula to find a volume with a hole.
- Find the volume of a solid of revolution using the Shell Method.
- Compare Shell and Disc Methods.
- Compute work done by a constant and variable force.
- Determine fluid pressure and fluid force as an integral application.
- Locate the moments of a mass.
- Locate the center of mass of a linear system and planar lamina.
- Find the centroid of a plane region.
- Calculate the arc length of various functions.
- Use arc length to find the surface area of a solid of revolution.
Techniques of Integration
- Review the basic integration formulas.
- Establish a procedure for fitting integrands to basis formulas.
- Introduce Integration by parts.
- Be able to use integration by parts a multiple amount of times.
- Integrate the trigonometric functions raised to powers.
- Manipulate integrals involving sine-cosine products with different angles.
- Introduce trigonometric substitution as a technique of integration.
- Use partial fractions to integrate rational functions (linear factors).
- Use partial fractions to integrate rational functions (quadratic factors).
- Be able to read and use tables of integrals.
- Use techniques of integration to establish reduction formulas.
- Substitute for rational functions of sine and cosine.
- Use L'Hopital's rule to determine indeterminate forms of limits.
- Do improper limits with infinite limits and integrands.
Infinite Series
- Introduce Taylor polynomials and approximations.
- Find MacLaurin polynomials.
- Approximate a functional value to a desired accuracy.
- Define sequence and limit of a sequence.
- Determine divergence/convergence of a given sequence.
- Define an infinite series, convergence and divergence.
- Establish the algebraic properties of infinite series.
- Formulate the nth-term test for divergent series.
- Create the formula for convergence of a geometric series.
- Introduce and use the Integral Test for infinite series.
- Find out when a p-series converges.
- Use the comparison test on infinite series.
- Establish and use the limit comparison test.
- Formulate the Alternating Series Test.
- Be able to discuss absolute and conditional convergence.
- Introduce the ratio and root tests for series.
- Summarize and process the various tests for convergence/divergence.
- Define Power series and their convergence.
- Find intervals of convergence of power series and test endpoints.
- Use calculus on power series.
- Be able to represent Transcendental Functions as power series.
- Define Taylor and MacLaurin series for composite functions.
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