Calculus II

What this covers

• Integration techniques: integration by parts (including repeated and cyclic cases), trigonometric integrals, trigonometric substitution, and partial fraction decomposition
• Improper integrals: infinite limits, discontinuous integrands, and comparison arguments for convergence
• Applications of the definite integral: volumes by disks, washers and cylindrical shells, arc length, surface area, work, and average value
• Sequences and infinite series: geometric and telescoping series, plus the divergence, integral, comparison, limit comparison, ratio, root, and alternating series tests
• Power series: radius and interval of convergence, endpoint testing, and term-by-term differentiation and integration
• Taylor and Maclaurin series: building series from known expansions, remainder estimation, and using series to approximate values and evaluate limits
• Parametric equations and polar coordinates: slopes, arc length, and area enclosed by polar curves

Where learners get stuck

Treating a sequence and its associated series as the same object — concluding a series converges because its terms approach zero

The nth-term test only works in one direction, and the notation a_n versus Σa_n looks similar on the page. The harmonic series is the standard counterexample, but students often meet it once and then forget it under time pressure.

Picking a convergence test by pattern-matching the first thing that looks familiar instead of by structure

Calc I rewarded a single correct procedure per problem type. Series problems have several valid routes and many invalid ones, so students need to read the terms for factorials, nth powers, or rational behaviour before committing — a habit that has to be built deliberately.

Setting up volume integrals with the wrong variable of integration or forgetting to subtract the inner radius

Washers and shells both come from rotating a region, and the formulas are memorised as symbols rather than as a picture of a slice. If you cannot state what one representative slice looks like and which axis it is perpendicular to, the setup collapses even when the integration is fine.

What a session looks like

You talk through problems out loud with Evelyn. A typical session might start with an integral you got stuck on: you read it aloud, and you are asked what structures you can see — a product, a radical of the form a² − x², a rational function — before any technique is named. For series, you are asked to justify the test you chose and state its hypotheses. Work is done on your own paper; you narrate steps and results, and Evelyn checks the reasoning and catches sign, bound, and substitution errors. Sessions can also be used to review a returned exam question by question, or to rehearse explaining a Taylor series derivation before a quiz.

Helpful to know first

• Derivatives, including chain rule, and implicit differentiation
• Definite and indefinite integrals, the Fundamental Theorem of Calculus, and u-substitution
• Limits, including L'Hôpital's rule and limits at infinity
• Algebra fluency: polynomial long division, factoring, and solving for partial fraction coefficients
• Trigonometric identities (Pythagorean, double-angle), inverse trig functions, and exponential and logarithm rules

Questions

Why is Calculus II considered harder than Calculus I?

Two reasons. Integration has no universal algorithm, so you must diagnose each problem and sometimes try a method that fails. And the second half of the course shifts from computation to convergence arguments, which require stating and checking hypotheses rather than executing steps.

Can voice tutoring work for math with this much notation?

You keep paper or a tablet beside you and do the writing. The conversation covers the decisions — which substitution, which test, which variable to integrate in — and you read your setup and answer back for checking. Long algebraic simplification is something you do between prompts rather than dictate line by line.

Do I have to memorise all the series convergence tests?

You need to recall the tests and their conditions, since most courses do not allow notes on that section. Sessions focus on the diagnostic order — check the nth-term test first, look for geometric or p-series form, then decide between comparison, ratio, or integral based on the shape of the terms.

My course covers parametric and polar but another section skips them. Can we match my syllabus?

Yes. Calc II content varies by institution — some include differential equations or centre of mass, others push polar curves into Calc III. Tell Evelyn which sections your course covers and which textbook you use, and sessions follow that ordering.