This course is the second installment of Single-Variable Calculus. The student will explore the mathematical applications of integration before delving into the second major topic of this course: series. The course will conclude with an introduction to differential equations. Upon successful completion of this course, students will be able to: Define and describe the indefinite integral; Compute elementary definite and indefinite integrals; Explain the relationship between the area problem and the indefinite integral; Use the midpoint, trapezoidal, and Simpson's rule to approximate the area under a curve; State the fundamental theorem of calculus; Use change of variables to compute more complicated integrals; Integrate transcendental, logarithmic, hyperbolic, and trigonometric functions; Find the area between two curves; Find the volumes of solids using ideas from geometry; Find the volumes of solids of revolution using disks, washers, and shells; Find the surface area of a solid of revolution; Compute the average value of a function; Use integrals to compute displacement, total distance traveled, moments, centers of mass, and work; Use integration by parts to compute definite integrals; Use trigonometric substitution to compute definite and indefinite integrals; Use the natural logarithm in substitutions to compute integrals; Integrate rational functions using the method of partial fractions; Compute improper integrals of both types; Graph and differentiate parametric equations; Convert between Cartesian and polar coordinates; Graph and differentiate equations in polar coordinates; Write and interpret a parameterization for a curve; Find the length of a curve described in Cartesian coordinates, described in polar coordinates, or described by a parameterization; Compute areas under curves described by polar coordinates; Define convergence and limits in the context of sequences and series; Find the limits of sequences and series; Discuss the convergence of the geometric and binomial series; Show the convergence of positive series using the comparison, integral, limit comparison, ratio, and root tests; Show the divergence of a positive series using the divergence test; Show the convergence of alternating series; Define absolute and conditional convergence; Show the absolute convergence of a series using the comparison, integral, limit comparison, ratio, and root tests; Manipulate power series algebraically; Differentiate and integrate power series; Compute Taylor and MacLaurin series; Recognize a first order differential equation; Recognize an initial value problem; Solve a first order ODE/IVP using separation of variables; Draw a slope field given an ODE; Use Euler's method to approximate solutions to basic ODE; Apply basic solution techniques for linear, first order ODE to problems involving exponential growth and decay, logistic growth, radioactive decay, compound interest, epidemiology, and Newton's Law of Cooling. (Mathematics 102; See also: Chemistry 004, Computer Science 104, Mechanical Engineering 002)
