![]() ![]() Moving to this new way of describing functions beyond elementary functions allows integration to beĪccomplished. That this function has no algebraic integral in terms of elementary functions (exponential, log, trig, roots,įinite polynomial, etc.) This leads to the concept of a function polynomial expansion - a way toĪpproximate a function by polynomials (bridging to equality when considering infinite polynomials). To a function, like f(x) = x * tan(x) ? The limitation is not a matter of cleverness - one can prove The second half of Calculus II explores the problem: What to do when you cannot find an antiderivative When Algebraic Integration Cannot Be Completed.Is common for this theorem to be explored only in Multivariable Calculus, but we do this theorem earlyĪs an introduction to the higher dimensional Fundamental Theorem of Calculus. In the latter case, the Gauss-Green Theorem is utilized it Varieties: over 2D regions that are essentially rectangular, and over 2D regions that are not rectangular,īut whose boundary curve can be formulated. Which can be used to measure volume and other applications. Double Integrals and Gauss-Green TheoremĬalculus II starts the dimensional generalization of integration theory, into double (or 2D) integrals,.With only manual tools - it is the leverage of computer algebra tools that makes this technique Presented, which is absent from all traditional textbooks, since it is computationally difficult For example,Ī more advanced integration technique known as Integration via Differentiation is We strive for aīalance between classical and modern computational mathematics in a unique way. The paper/pencil standpoint, which has merits and drawbacks in this modern age. Traditional Calculus II courses explore these techniques purely from One of the goals of Calculus II is to become an expert in algebraic integration: finding antiderivatives.Ĭomputer algebra tools can find antiderivatives automagically, so an exploration of the techniques ofĪntiderivatives must contain an meaningful mixture of integration concepts, manual skills, and usage Important for all students to have this common foundation moving forward into the Calculus II course. ![]() Integration, the Fundamental Theorem of Calculus, and initial applications of the integral.Īs many students start in our Calculus II course, having studied Calculus I elsewhere, it is Starting from the definition of the integral via signed area, ranging to beginning algebraic Our Calculus II course has the following components:Ĭalculus II starts with an intensive 40 assignment refresher of introductory integral calculus, Many students feel that Calculus II is the most difficult course in the Calculus sequence as well. This raises all kinds of questions that have toīe studied, but once accomplished, we are able to conquer these algebraicĬalculus II is the longest course in the Calculus sequence. but with a more generalized description based upon "Plan B" for attacking these types of algebraic integrals comes in the form ofĮxpanding the way we describe functions, not just with the elementaryĬlass of functions including such friends as sin(x), e x, Just is no algebraic antiderivative for such functions. The vast majority of functions cannot be algebraically integrated - there
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