bythreedu · Math Tracks · Curriculum v1.0 · 2026

Integration techniques,
series, and the mathematics
of
convergence.

A complete Calculus II course covering integration techniques, improper integrals, geometric applications, sequences and series, power series, Taylor series, and an introduction to parametric and polar. Built for students who finished Calc I and are ready to go deeper.

Duration~18weeks
Total Lessons19at your pace
Units6progressive phases
Assignments6+problem sets
DestinationCalc III Ready
00Approach
Course Philosophy

Integration Is a Craft

Differentiation has rules you can apply mechanically. Integration doesn't — it requires recognizing structure and choosing a technique. This course builds that recognition skill through deliberate practice, not pattern matching to a formula sheet.

Series Are Not Optional

Most Calc II students coast through integration and then hit a wall with series. This course treats series as a core topic, not an appendix. The convergence tests, power series, and Taylor expansions are taught at the same depth as integration.

01Curriculum
Learning Path
01
Unit 1 · Weeks 1–3
Integration Techniques
4 lessons · U-substitution · Integration by parts · Trig integrals · Partial fractions
u-substitutionintegration by partspartial fractions
+
L1
U-Substitution
Recognizing the composite structure. Choosing u correctly. Definite integrals with substitution — changing the limits.
L2
Integration by Parts
Derived from the product rule. The LIATE heuristic for choosing u and dv. Tabular integration for repeated application.
L3
Trig Integrals and Trig Substitution
Powers of sin and cos. Trig substitution for integrals involving √(a²-x²), √(a²+x²), and √(x²-a²). Completing the square first.
L4
Partial Fraction Decomposition
Breaking rational functions into partial fractions. Distinct linear, repeated linear, and irreducible quadratic factors. Integration after decomposition.
Assignment 1
Evaluate 20 integrals — 5 each of u-substitution, IBP, trig integrals, and partial fractions. Label which method you used and why.
You evaluate any standard integral by identifying the right technique.
02
Unit 2 · Weeks 4–5
Improper Integrals and Applications
3 lessons · Improper integrals · Area between curves · Arc length
improper integralsconvergencearea between curves
+
L5
Improper Integrals
Infinite limits of integration and integrands with vertical asymptotes. Convergence and divergence. Comparison theorem for convergence without computing.
L6
Area Between Curves
Setting up the integral correctly — which function is on top and on what interval. Integrating with respect to y when that is simpler.
L7
Arc Length and Surface Area
Arc length formula derived from the Pythagorean theorem applied to infinitesimal segments. Surface area of revolution.
Assignment 2
Evaluate 5 improper integrals (classifying convergent or divergent). Set up and evaluate 4 area between curves problems showing the setup sketch.
You evaluate improper integrals and set up geometric applications correctly.
03
Unit 3 · Weeks 6–8
Volumes of Solids
3 lessons · Disk and washer · Shell method · Method selection
disk methodwasher methodshell method
+
L8
Disk and Washer Method
Solids of revolution around the x-axis and y-axis. When to use disk (no hole) vs. washer (with hole). Setting up the integral from the geometry.
L9
Shell Method
Cylindrical shells as an alternative to washers. When shells are simpler: rotating around a vertical axis but integrating with respect to x.
L10
Choosing the Right Method
The decision: which axis of rotation, which variable of integration, which method? Making the choice before starting so the setup is correct.
Assignment 3
Set up and evaluate 6 volume problems — 2 disk, 2 washer, 2 shell. For each, draw the region and the solid before writing the integral.
You set up any volume of revolution integral and choose the most efficient method.
04
Unit 4 · Weeks 9–11
Sequences and Series
3 lessons · Sequences · Series convergence · Convergence tests
convergence testsgeometric seriesp-series
+
L11
Sequences
Limits of sequences. Monotone and bounded sequences. The Monotone Convergence Theorem as intuition.
L12
Series and Basic Tests
Geometric series sum formula. Telescoping series. The divergence test — what it can and cannot tell you.
L13
Convergence Tests
Integral test, comparison test, limit comparison, ratio test, root test, alternating series test. Choosing which test to apply and why.
Assignment 4
Test 15 series for convergence. For each, identify which test you used and what made it the right choice. Document what each test actually tells you.
You determine whether any standard series converges or diverges.
05
Unit 5 · Weeks 12–14
Power Series and Taylor Series
3 lessons · Power series · Taylor series · Applications
radius of convergenceTaylor seriesMaclaurin series
+
L14
Power Series
Radius and interval of convergence. Differentiating and integrating power series. Representing functions as power series.
L15
Taylor and Maclaurin Series
Deriving the Taylor series from the requirement that derivatives match. The key series: e^x, sin x, cos x, ln(1+x), 1/(1-x). Taylor's remainder theorem.
L16
Applications of Series
Approximating integrals using series. Evaluating limits using series expansion. Binomial series. Error bounds for approximations.
Assignment 5
Derive the Maclaurin series for e^x and sin x from scratch. Then use series to evaluate 3 limits and approximate 2 integrals.
You work with power series and Taylor expansions as tools, not formulas to look up.
06
Unit 6 · Weeks 15–18
Parametric Equations, Polar Coordinates, and Capstone
3 lessons + capstone
parametric derivativespolar areaconic sections in polar
+
L17
Parametric Equations
Derivatives of parametric curves. Arc length in parametric form. Eliminating the parameter vs. keeping it.
L18
Polar Coordinates
Converting between rectangular and polar. Derivatives and arc length in polar. Area enclosed by polar curves.
L19 (cap)
Capstone: Simulated Final Exam
A full-length Calc II exam simulation covering all 6 units. Every missed problem corrected with full reasoning.
Assignment 6
A full-length simulated Calc II final. Every missed problem gets a written correction with the reasoning that should have driven each step.
You complete Calculus II ready for multivariable calculus or any application that builds on it.
02Reference
Core Concepts
∫u dv
Integration by Parts
Derived from the product rule: ∫u dv = uv - ∫v du. Choose u using LIATE (logarithm, inverse trig, algebraic, trig, exponential). The technique reduces one integral to another.
U-substitution · Trig substitution · Partial fractions
∞
Improper Integrals
An improper integral has an infinite limit of integration or an integrand with a vertical asymptote. It is defined as a limit of proper integrals. It may converge to a finite value or diverge.
Comparison theorem · Convergence · p-integrals
Σ
Series Convergence
A series converges if its sequence of partial sums approaches a finite limit. The divergence test can only prove divergence. Every other test is needed to confirm convergence.
Geometric series · p-series · Ratio test
R
Radius of Convergence
A power series converges absolutely inside its radius of convergence and diverges outside it. At the endpoints, the series may converge or diverge — each endpoint must be tested separately.
Interval of convergence · Taylor series · Ratio test
e^x = Σx^n/n!
Taylor Series
A Taylor series represents a function as an infinite polynomial. The coefficients are determined by the function's derivatives at a point. The key series — e^x, sin x, cos x — should be known from memory.
Maclaurin series · Remainder · Approximation
r(θ)
Polar Coordinates
A point in polar coordinates is given by (r, θ) — distance from the origin and angle from the positive x-axis. Area in polar: A = (1/2)∫r² dθ.
Parametric equations · Arc length · Polar area
03Capstone
Simulated Final Exam

The capstone is a full-length Calc II exam simulation covering every unit. Every missed problem gets a written correction that identifies the decision that should have been made — not just the right answer.

○
Gap Review
Two weakest units revisited before the exam.
○
Full Exam Simulation
All 6 units, timed, independent.
○
Written Corrections
Every miss corrected with reasoning.
○
Calc III Readiness
A final session on what multivariable calculus will require.
04Investment
Pricing Options
per Hour
$85
1-hr lesson
Flexible · No commitment
Online only
  • Flexible scheduling
  • 1-on-1 focused attention
  • No long-term commitment

Best for trying the curriculum before committing.

Book a session
Popular
per Month
$650
4 lessons · 1.5 hours/lesson
1 lesson/week · book more to move faster
Online + In-Person available
  • Session notes after every lesson
  • Async DM support between sessions
  • Priority scheduling

Booking 2x/week draws down lessons faster — no extra charge.

Get started
Semester
$3,775
26 lessons · 1.5 hours/lesson
~6 months · save vs. monthly
Online + In-Person available
  • Full course arc
  • Session notes after every lesson
  • Capstone review + corrections
  • Next track readiness session

The full arc in one commitment.

Enroll now
In-person sessions available on Monthly and Semester plans at $800/mo and $4,625/semester. NYC only. Per-hour sessions are online only.
05Contact
Let's Talk

Moving into Calc II from Calc I or need a structured second pass at it? One session is enough to figure out where to start.

What do I need coming in? +
Solid Calc I: limits, differentiation rules including chain rule, and the Fundamental Theorem of Calculus. If any of those are shaky, the Calculus I track addresses them.
Is BC Calculus covered here? +
Yes. The Calc II track covers the BC topics beyond AB: integration techniques, series, polar and parametric.
Are lessons online or in-person? +
Per-hour sessions are online only. Monthly and Semester plans include the option for in-person sessions in NYC.
What if I need to reschedule? +
Give 24 hours notice and we'll find another time. Monthly and Semester students get priority rebooking.