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Calculus II course

Develop integration methods, sequences, series, and parametric reasoning.

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7Core units
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Units at a glance

Every unit in this course, what you will be able to do once you finish it, and roughly when it lands in a 14-week schedule.

Calculus II course map: 7 units
#UnitWhat you can do after itWeeks
01Integration techniquesYou can pick substitution, parts, partial fractions or a trig method for a hard integral.Weeks 1-2
02Applications of integrationYou can use integrals for area between curves, volume of a rotated shape, arc length and average value.Weeks 3-4
03Improper integralsYou can integrate over an endless range, or across a point where the function blows up, by using a limit.Weeks 5-6
04SequencesYou can decide whether a list of numbers settles down to a limit or wanders off.Weeks 7-8
05Infinite seriesYou can decide whether an endless sum reaches a finite total, using the test that fits.Weeks 9-10
06Power seriesYou can write a function as an endless polynomial and state the x values where it works.Weeks 11-12
07Parametric and polar curvesYou can describe a path with a time parameter, or with an angle and a radius, and find slopes and areas from it.Weeks 13-14

Calculus II course units

Work in order for a complete course, or jump directly to the unit you need for an upcoming assessment.

01

Integration techniques

Weeks 1-2

You can pick substitution, parts, partial fractions or a trig method for a hard integral. You can also check your answer by differentiating it back.

  • Choose u so that du already appears in the integral.
  • Use integration by parts on x times e to the x.
  • Split a fraction into partial fractions before integrating.

Worked example

Integrate 2x(x squared + 1) cubed dx

  1. Let u = x squared + 1, so du = 2x dx
  2. The integral becomes u cubed du, which is u to the fourth over 4

Answer(x squared + 1) to the fourth, divided by 4, plus C

Most common mistakeChanging the variable but not the differential: writing the integral of u cubed dx leaves a stray 2x that never gets accounted for.

Explain integration techniquesPractice integration techniques
02

Applications of integration

Weeks 3-4

You can use integrals for area between curves, volume of a rotated shape, arc length and average value. You can also choose whether to slice vertically or horizontally.

  • Set up the area between y = x and y = x squared from 0 to 1.
  • Use the disk method for a region spun about the x-axis.
  • Find the average value of a function on an interval.

Worked example

Find the area between y = x and y = x squared from x = 0 to x = 1

  1. On this interval y = x is on top, so integrate x - x squared
  2. x squared / 2 - x cubed / 3, evaluated from 0 to 1: 1/2 - 1/3

Answer1/6

Most common mistakeSubtracting in the wrong order: integrating x squared - x gives -1/6, a negative area for a region that plainly sits above the axis.

Explain applications of integrationPractice applications of integration
03

Improper integrals

Weeks 5-6

You can integrate over an endless range, or across a point where the function blows up, by using a limit. You can also say whether the answer is a finite number.

  • Rewrite an infinite upper limit as a limit with a bound b.
  • Decide convergence for 1 over x to the p by comparing p with 1.
  • Split an integral at an interior point where the function blows up.

Worked example

Evaluate the integral of 1/x squared from 1 to infinity

  1. Antiderivative is -1/x
  2. Take the limit of (-1/b + 1) as b grows

Answer1

Most common mistakeSubstituting infinity directly instead of taking a limit: the same careless move on the integral of 1/x from 1 to infinity hides that it diverges.

Explain improper integralsPractice improper integrals
04

Sequences

Weeks 7-8

You can decide whether a list of numbers settles down to a limit or wanders off. You can also compare growth rates to find that limit quickly.

  • Find the limit of (3n + 1)/(n + 5) as n grows.
  • Argue convergence from a sequence being increasing and bounded above.
  • Use the fact that r to the n goes to 0 when r is between -1 and 1.

Worked example

Find the limit of a(n) = (3n + 1)/(n + 5)

  1. Divide top and bottom by n: (3 + 1/n)/(1 + 5/n)
  2. Both 1/n and 5/n shrink to 0

Answer3

Most common mistakeReading the limit off the constants: answering 1/5 = 0.2, which ignores that the n terms dominate everything once n is large.

Explain sequencesPractice sequences
05

Infinite series

Weeks 9-10

You can decide whether an endless sum reaches a finite total, using the test that fits. You can also add a geometric series exactly.

  • Apply the ratio test to a series containing factorials.
  • Use the p-series rule on sums of 1 over n to a power.
  • Sum a geometric series with a over (1 - r).

Worked example

Does the sum of 1/n squared converge? What about the sum of 1/n?

  1. Both have terms shrinking to 0, which settles nothing
  2. p-series rule: converges only when p is bigger than 1

Answer1/n squared converges (p = 2); 1/n diverges (p = 1)

Most common mistakeConcluding a series converges because its terms shrink to zero: the harmonic series 1 + 1/2 + 1/3 + ... has shrinking terms and still grows past every bound.

Explain infinite seriesPractice infinite series
06

Power series

Weeks 11-12

You can write a function as an endless polynomial and state the x values where it works. You can also use the first few terms to approximate a number.

  • Find the radius of convergence with the ratio test.
  • Write the series for e to the x out to the cubic term.
  • Estimate e to the 0.1 with three terms and bound the error.

Worked example

Estimate e to the power 0.1 using 1 + x + x squared / 2

  1. 1 + 0.1 + 0.005
  2. = 1.105

Answer1.105, against a true value of 1.10517

Most common mistakeUsing a series outside its interval: putting x = 3 into 1 + x + x squared + ... for 1/(1 - x) gives a runaway sum for a value that is really -0.5.

Explain power seriesPractice power series
07

Parametric and polar curves

Weeks 13-14

You can describe a path with a time parameter, or with an angle and a radius, and find slopes and areas from it. You can also convert between those forms and x-y form.

  • Find dy/dx for x = t squared, y = t cubed using (dy/dt) over (dx/dt).
  • Convert the polar point with r = 2 and angle pi/3 into x and y.
  • Find the area inside r = 2 cos of the angle with the polar area integral.

Worked example

For x = t squared and y = t cubed, find dy/dx at t = 2

  1. dy/dt = 3t squared and dx/dt = 2t
  2. dy/dx = 3t squared / 2t = 3t/2

AnswerAt t = 2 the slope is 3

Most common mistakeDividing the functions rather than their derivatives: computing t cubed over t squared = 2 as the slope instead of 3.

Explain parametric and polar curvesPractice parametric and polar curves

Prepare for Calculus II practice

Start with the earliest uncertain prerequisite

Before timing yourself, check whether you can explain Integration techniques from a blank page. Then connect it to Applications of integration. If either explanation depends on copying a formula, review the unit first and complete two untimed examples.

Use tools to verify, not to choose the method for you

The Calculus calculator can test calculations and representations used in Calculus II. Make the setup yourself, predict the sign or scale, and compare the tool result with that prediction. Use the formula library to check conditions as well as notation.

Know when to move to the full test

Move from Calculus II practice to the complete course test after you can correct a missed problem without reopening the worked answer. Record the earliest wrong decision—not only the final score—so the next study session has a precise target.

Before and after the syllabus

Learn the ideas, then practise them

The unit list tells you what is covered. These two pages are where the method is explained and where you find out whether it stuck.

GuidesCalculus explainedConcepts, worked methods and checks behind this course. PracticeMixed Calculus questionsAnswer review after every attempt. ReferenceFormulas for this courseConditions and a worked example for each one.
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Assess

Take the complete Calculus II test

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Review essential formulas

Open library →Calculate

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Prepare around your exam date

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Questions about the Calculus II course

Where should I start?

Start with Integration techniques if you are following the full sequence. If that unit feels automatic, open the Calculus II practice page, choose mixed review, and let the first errors identify the earliest prerequisite to revisit.

How do I know I am ready for the course test?

You are ready when you can choose a method without a hint, show the governing steps, and explain why the result is reasonable. Use the complete Calculus II test only after you can correct practice errors from a blank page.

Which calculator supports this course?

The Calculus calculator supports the calculations and representations used in this course. Use it to test or visualize a result after making your own setup.