**Course** Description: | A more rigorous and extensive treatment of integral calculus. Topics include the Fundamental Theorem of calculus, applications of integration, logarithms and exponentials, Taylor polynomials, sequences, series, and uniform convergence. |

**Course Objectives:** | n addition to learning the customary computational skills in calculus,
the student will be exposed to mathematics as mathematicians view it.
Students will learn to write proofs and think more rigorously about
mathematics, and will come to grips with challenging concepts and problems.
In that regard, this course is an excellent preparation for students considering
a career in law, medicine, or the sciences. |

**Topical Outline:** | 1. Upper and lower sums, the definition of the integral, the convenient criterion
for integrability; examples and counterexamples.
2. Properties of the integral.
3. The Fundamental Theorem of Calculus.
4. Applications: areas under curves, volumes, arclength, work.
5. The logarithm and exponential functions.
6. Methods of integration: integration by substitution, by parts, by partial
fractions (including sketch of proof), by trigonometric substitution. Improper integrals.
7. Taylor polynomials; in-depth treatment of the algebra and calculus of
Taylor polynomials. Taylor's Theorem with remainder, with applications to
indeterminate forms and approximate integration.
8. Sequences and series. Bolzano-Weierstrass Theorem, Cauchy sequences.
Comparison and limit comparison tests, ratio test, integral test. Conditional
convergence and rearrangement.
9. Sequences and series of functions. Pointwise and uniform convergence.
Applications to power series and computation of explicit numerical series.
10. Complex numbers and power series. Explanation of singularities on the circle of
convergence. |