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∫ Calculus II

Continue into integration and its many uses. You'll evaluate integrals, compute areas and volumes, and work with infinite series and convergence.

15
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~90 min
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🔢 Math
subject
Adults
level
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What you’ll learn

  1. The Two Problems of CalculusFrame Calculus II as the study of accumulation and place it in its Newton-Leibniz historical context.Integral calculus answers the inverse of Calculus I's question: given a rate, find the total. Newton and Leibniz discovered the connection independently in the 17th century, Leibniz's ∫ and dx notation won, and rigor arrived with Cauchy and Riemann in the 19th.
  2. The Integral as AccumulationDefine the definite integral as a limit of Riemann sums and internalize the slice-sum-limit pattern.The integral accumulates a rate into a total, the way an odometer accumulates the speedometer. Riemann's 1854 definition — rectangle sums with widths shrinking to zero — grounds it, and the slice-sum-limit recipe generates every application formula in the course.
  3. The Fundamental TheoremState and use both forms of the Fundamental Theorem of Calculus.∫ₐᵇ f dx = F(b) − F(a) for any antiderivative F, collapsing infinite sums to a subtraction; conversely the accumulation function A(x) satisfies A′ = f. It converts Archimedes-level feats into routine computation.
  4. SubstitutionEvaluate integrals by u-substitution, including converting limits in definite integrals.Substitution reverses the chain rule: pick u as the inner function whose derivative appears alongside, rewrite entirely in u, integrate, and substitute back — or convert the limits and stay in u for definite integrals.
  5. Integration by PartsApply integration by parts with sound u/dv choices guided by LIATE.From the product rule comes ∫u dv = uv − ∫v du: trade the given integral for a simpler one by differentiating the factor that simplifies (logs, inverse trig, polynomials) and integrating the one that doesn't mind (exponentials, trig).
  6. Trig Integrals and SubstitutionsIntegrate powers of trig functions and use trig substitution on radical integrands.Odd trig powers peel off a factor for du; even powers step down via half-angle identities. Radicals √(a²±x²), √(x²−a²) dissolve under the sine, tangent, and secant substitutions — the algebraic signature of hidden circles.
  7. Partial FractionsDecompose rational functions and integrate the resulting simple pieces.Factor the denominator, write the template (one term per power of each factor, linear numerators over quadratics), solve for constants, and integrate to logarithms and arctangents — dividing first if the fraction is improper.
  8. Improper IntegralsEvaluate integrals over infinite intervals or with unbounded integrands as limits, using the p-test.Cauchy's definition makes improper integrals limits of proper ones: ∫₁^∞ dx/xᵖ converges iff p > 1, endpoint spikes can still enclose finite area, and Torricelli's trumpet shows finite volume can coexist with infinite surface.
  9. Area Between CurvesCompute areas between curves by integrating top-minus-bottom between intersections.Find the crossings, determine which curve is on top, and integrate the gap — splitting where the curves trade places. The same 'integrate the gap' pattern powers engineering quantities from dam forces to airfoil lift.
  10. Volumes by SlicingCompute volumes by the disk and washer methods.Volume is the integral of cross-sectional area: V = ∫A(x) dx, with disks π r² (washers subtract the hole) for solids of revolution. The method derives the sphere's 4πr³/3 in four lines and describes every lathe-turned object.
  11. Volumes by ShellsCompute volumes with cylindrical shells and choose wisely between disks and shells.Slices parallel to the axis give nested shells contributing 2πx·f(x) dx, often avoiding the need to invert functions. Kepler's 1615 wine-barrel volumes pioneered the slicing idea decades before formal calculus.
  12. Arc Length and WorkSet up and evaluate integrals for curve length and for work done by varying forces.Arc length sums Pythagorean pieces: ∫√(1+f′²) dx — the computation behind bridge cables and the Gateway Arch's catenary. Work integrates force over distance, layer by layer for pumping problems, and a water tower is that integral held in reserve.
  13. Sequences and LimitsDefine sequence convergence and evaluate standard limits, including the definition of e.A sequence converges when its terms eventually stay arbitrarily close to one limit. Key instincts: n/(n+1) → 1, oscillators diverge, exponentials beat polynomials, and (1+1/n)ⁿ → e, Euler's compounding constant.
  14. Infinite SeriesDefine series convergence via partial sums; master geometric and p-series and the divergence warning.A series converges when its partial sums do: geometric series sum to a/(1−r) for |r| < 1, the harmonic series diverges despite vanishing terms, and Σ1/nᵖ needs p > 1. Euler's Basel solution Σ1/n² = π²/6 crowned the subject.
  15. Taylor SeriesConstruct Taylor and Maclaurin series and use polynomial truncations as approximations.Taylor's 1715 expansion rebuilds smooth functions from their derivatives at a point; partial sums approximate with growing fidelity, which is how calculators evaluate eˣ, sin, and cos. Knowing five basic series — and the small-x habits sin x ≈ x — pays forever.

Questions this course answers

The core question of integral calculus is:

Differential calculus studies rates; integral calculus accumulates them into totals — and the Fundamental Theorem says the two are inverse operations.

The ∫ symbol comes from:

Leibniz designed the notation — ∫ as a stretched S for sum, paired with the differential dx — and its usability is a big reason it displaced Newton's notation.

A Riemann sum approximates a definite integral by:

Slice, sum, take the limit: the integral is defined as the limit of rectangle sums as the slice width goes to zero.

By the Fundamental Theorem, ∫₀¹ x² dx equals:

An antiderivative of x² is x³/3; evaluating from 0 to 1 gives 1/3 − 0 = 1/3.

If A(x) is the accumulated area under f from a to x, then A′(x) equals:

The area function grows exactly at the height of the curve: A′(x) = f(x). That inverse relationship is the theorem's second form.

For ∫ 2x cos(x²) dx, the winning substitution is:

u = x² gives du = 2x dx, which absorbs the 2x factor; the integral becomes ∫cos u du = sin(x²) + C.

Grounded in trusted sources

  • James Stewart — Calculus: Early Transcendentals (8th ed., 2015)
  • Gilbert Strang & Edwin Herman — Calculus Volume 2 (OpenStax, 2016)
  • C.H. Edwards — The Historical Development of the Calculus (1979)
  • William Dunham — Journey Through Genius: The Great Theorems of Mathematics (1990)
  • MacTutor History of Mathematics Archive (University of St Andrews)

Every Wunder lesson is built from real, reputable sources — never invented.

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