Calculus 2 Unlocked

Explore the disk and washer method to compute volume by slicing and summing infinitesimal disks, using outer minus inner radii and careful boundary setup.

Learn to compute volumes with the shell method, slicing parallel to the rotation axis, using circumference and height, with y axis rotation examples.

Explore calculating work with integrals through slicing and scooting problems—lifting ropes, pumping water, leaky buckets, and springs—using weight, density, and gravity with proper units (ft, lb, joules).

Compute the average value of a function by integrating from a to b and dividing by b minus a, and apply the mean value theorem: f(c) equals the average.

Apply integration by parts using ∫ u dv = uv − ∫ v du, choose u by the L U A T E rule, and resolve loops in tricky integrals.

Master solving trigonometric integrals by applying common identities, recognizing tangent and secant shortcuts, and using substitution for odd/even power cases.

Master trigonometric substitution to evaluate integrals, distinguish it from u-substitution, pick sine, secant, or tangent paths, use triangles to map theta to x, and apply identities.

Explore partial fraction decomposition, including factoring, long division, and decomposing into distinct linear, repeated, and irreducible quadratic forms to prepare for integrating rational functions.

Explore improper integrals by turning infinite bounds into limits, handling discontinuities and domain checks, and testing convergence or divergence while carrying the limit to the end.

Learn to compute arc length using the formula sqrt(1+(f'(x))^2) dx from a to b, verifying domain and continuity, and applying tricks like completing the square or trig substitution.

Learn to compute the surface area of a surface of revolution by slicing it into belts and integrating, using the 2π f(x) sqrt(1+f'(x)^2) formula (and similar for the y-axis).

Explore the application of integrals to continuous probability distributions, including the exponential distribution for waiting times, and compute mean and median using density functions and area constraints.

Explore sequences in calculus two, including sequence notation and formulas, domain natural numbers, limit laws, and L'Hôpital's rule, to determine convergence or divergence with practical examples.

Explore basics of series as the infinite sum of a sequence, with notation from n equals one to infinity, and the ideas of partial sums and convergent or divergent series.

Explore the divergence test for infinite series, determining divergence from the limit of the sequence. A nonzero limit implies divergence; zero implies inconclusive, requiring other tests.

Explore geometric series, derive the partial sum formula, and apply convergence criteria to determine sums for series with ratio r, including index handling.

this lecture explains applying the integral test to improper integrals by verifying positivity, continuity, and decreasing behavior, then comparing series with corresponding integrals to decide convergence or divergence.

Compare a series to a known convergent or divergent benchmark, such as the harmonic series, using positive terms and limit comparison when the basic test is inconclusive.

Use the ratio test by substituting n+1 for n and taking the limit of a_{n+1}/a_n in absolute value: if < 1, converges; if > 1, diverges; if = 1, inconclusive.

Apply root test by taking the limit of the nth root of the absolute value; if the limit < 1, the series converges absolutely; if > 1, it diverges.

Explore alternating series in calculus 2, applying the alternating series test, divergence test, and criteria for decreasing terms, zero limit, and absolute vs conditional convergence, with practical error estimation.

Explore power series, learn to find their radius and interval of convergence using ratio and root tests, and test endpoints for absolute or conditional convergence.

Represent functions as power series by rewriting them in a geometric form a/(1-r), then differentiate or integrate term by term and find the radius and the interval of convergence.

Explore how to express a function as a Taylor or Maclaurin series, build the Taylor polynomial, and use derivatives at the center to approximate functions like e^x, sine, and cosine.