Integral calculus, applications of the integral, parametric curves and polar coordinates, power series and Taylor series.
Prerequisite: Minimum of C- in MATH 165 or high math placement scores
Credit hours: 4
An exploration of advanced techniques of integration. Applications of integration, including geometry, physics, polar coordinates and parametric curves, are explored. Power series are introduced and applications of approximating functions are given.
The etext version of Pearson’s textbook Thomas’ Calculus, Early Transcendentals, 14th Edition, by George Thomas, Jr., Maurice D. Weir and Joel Hass accompanies a purchased access code. The access code provides access to the online homework package. Students should access the online homework (MyMathLab) only through Canvas.
Application of Integration (Sections 6.1-6.6, 11.1-11.5) includes discussion of calculating volume using cross-sections and cylindrical shells, arc length, areas of surfaces of revolution, work and fluid force, moments and centers of mass, calculus with parametric curves and polar graphs.
Advanced Integration Techniques (Sections 8.1-8.5, 8.7, 8.8) includes discussion of integration by parts, trigonometric integrals, trigonometric substitution, integration of rational functions using partial fraction decomposition, numerical integration, and improper integration.
Infinite Sequences and Series (Sections 10.1 – 10.10) includes discussion of sequences, series, tests for convergence, power series, intervals of convergence, Taylor series, and applications of series.
After completing Math 166, students should be able to:
- Find the volume of a 3-dimensional solid, by drawing and labeling the 2-dimensional cross-section. Apply basic geometry to determine the volume of a “slice,” and set up the definite integral.
- Find the volume of a 3-dimensional solid by revolution using cylindrical shells.
- Calculate the length of a curve.
- Find the area of a surface of revolution.
- Calculate the amount of work done in stretching or compressing a spring or in moving a mass along a horizontal or vertical path.
- Calculate the fluid force acting on an object like the side of a tank.
- Calculate the moments, centers of mass, and centroids of thin wires and thin flat plates.
- Convert a pair of parametric equations to a Cartesian representation of an equation or from a Cartesian representation to parametric form.
- Find the equation of the line tangent to a curve at a given point, given a pair of parametric equations for a curve.
- Find the area of a region bounded by one or more parametric curves and find the length of a curve given in parametric form.
- Convert equations given in Cartesian form to equivalent expressions in polar coordinates, and convert equations given in polar form to equivalent expressions in Cartesian form.
- Plot points and equations in polar coordinates.
- Express equations given in polar coordinates in parametric equation form.
- Find the (Cartesian) equation of a line tangent to a curve described by a polar equation.
- Find the area bounded by one or more curves described by polar equations.
- Find the length of curves described by polar equations.
- Solve integrals using integration by parts.
- Solve integrals involving powers of trigonometric functions.
- Evaluate integrals by applying an appropriate trigonometric substitution and evaluating the resulting trigonometric integral.
- Given a rational function, rewrite the integrand as needed using partial fraction decomposition and proceeding with other techniques of integration.
- Apply numerical integration techniques (including left or right endpoint, midpoint, trapezoidal rule, or Simpson’s rule) to approximate a definite integral.
- Estimate the error involved with using a numerical integration technique when trapezoidal or Simpson’s rules are applied.
- Given an improper integral, determine whether it converges or diverges by evaluation, the direct comparison test, or the limit comparison test.
- Evaluate the limit of a sequence, if one exists. This may require application of L’Hopital’s Rule.
- Explain what it means for an infinite series to converge or diverge.
- Find the sum of a converging geometric series or identify when a geometric series diverges.
- Find the sum of a telescoping series or determine that the limit does not exist.
- Apply various tests to determine convergence or divergence of an infinite series. Tests include the nth term test, integral test, comparison tests, ratio test, root test, and alternating series test.
- Find the radius and interval of convergence of a power series.
- Write a Taylor or Maclaurin series or polynomial for a function given a center.
- Derive the Maclaurin series for sin(x), cos(x), e^x, 1/(1-x), arctan(x), and ln(1+x).
- Determine the nth order Taylor polynomial, P_n(x;a), about a specified center a, and calculate a corresponding error approximation, R_n(x;a) on a given interval centered at a.
- Develop the binomial series for expressions taking the form (1+x^k)^m.
- Use power (Taylor) series to approximate definite integrals of non-elementary functions and to evaluate limits taking indeterminate forms.
Three midterm night exams and a cumulative final exam are given. Times for the night exams may be found at the night exam schedule at https://www.registrar.iastate.edu/students/exams.
Weekly quizzes are administered in recitation session.
Three online (MyMathLab) homework assignments are due most weeks. The online homework package with a purchased access code includes the electronic textbook.
Many materials, including videos, practice exams, and quizzes, are available at calc2.org.
Math 166 help
Math 166 students may access the help hour times and location by following the link established in the Canvas course pages.