Integration techniques

5 lessons

Integration by parts, trigonometric integrals and substitutions, partial fractions, and choosing among them.

Review first: u-substitution Dividing before integrating Completing the square inside an integral

1.1Integration by parts

1.2Trigonometric integrals

1.3Trigonometric substitution

1.4Integration by partial fractions

1.5Choosing an integration technique

Improper integrals

2 lessons

Integrals over infinite intervals or across a blow-up, and when they add up to a finite number.

Review first: Limits at infinity and horizontal asymptotes Comparing growth rates at infinity

2.1Improper integrals

2.2Improper integrals with an infinite integrand

Further applications: arc length and more

4 lessons

Arc length, surface area, the shell method and work, each built as a limit of sums.

Review first: The disk method The washer method

3.1Arc length

3.2Surface area of a solid of revolution

3.3The shell method

3.4Work as an integral

Differential equations: logistic growth and Euler's method

2 lessons

Stepping along a slope field to approximate a solution, and growth with a ceiling.

Review first: Slope fields Separable differential equations

4.1Euler's method

4.2Logistic growth

Sequences and series: convergence

10 lessons

When an infinite sum has a finite value, and the tests that decide it.

Review first: Arithmetic series and partial sums Infinite geometric series

5.1Convergence of sequences

5.2Infinite series and partial sums

5.3The geometric series test

5.4The nth term test for divergence

5.5The integral test and p series

5.6The direct and limit comparison tests

5.7Alternating series and their error bound

5.8Absolute and conditional convergence

5.9The ratio and root tests

5.10Choosing a convergence test

Power series and Taylor series

7 lessons

Writing functions as polynomials of infinite degree, where those series converge, and how close a partial sum is.

6.1Taylor polynomials

6.2Power series and the interval of convergence

6.3Representing functions as power series

6.4The common Maclaurin series

6.5The Taylor remainder and the Lagrange error bound

6.6The binomial series

6.7Using series to evaluate limits and integrals

Parametric, polar and vector-valued functions

5 lessons

Derivatives, speed, arc length and area for curves given by parameters, polar equations and vectors.

Review first: Parametric equations Converting between polar and rectangular Vector addition and scalar multiplication

7.1Derivatives of parametric curves

7.2Arc length and speed on parametric and polar curves

7.3Vector valued functions and motion in the plane

7.4Slope of a polar curve

7.5Area in polar coordinates

Calculus 2