Glossary

\sum a_n converges ABSOLUTELY if \sum|a_n| converges too; CONDITIONALLY if \sum a_n converges but \sum|a_n| doesn't.

A function g(x)=\int_a^x f(t)\,dt that reports the running total of f's signed area up to x.

A series like \sum(-1)^n b_n converges if b_n\ge0 decreases to 0.

|S-S_N|\le b_{N+1} — the error from stopping early is at most the size of the FIRST omitted term.

A function F whose derivative is the given function f — reversing differentiation.

L=\int_a^b\sqrt{1+[f'(x)]^2}\,dx — the exact length traced by a curve y=f(x).

\int_a^b\left[\text{top}(x)-\text{bottom}(x)\right]dx — always subtract the LOWER curve from the upper one.

A differential equation y'=f(y) (no explicit t); an equilibrium is where f(y)=0, and can be stable or unstable.

\dfrac{1}{b-a}\int_a^b f(x)\,dx — the height of the rectangle with the same area as the curve.

(1+x)^k=\sum\binom{k}{n}x^n — the Maclaurin series generalizing the Binomial Theorem to any exponent k, not just positive integers.

Substituting into, adding, or scaling a known Maclaurin series, rather than differentiating from scratch.

The region's geometric "balance point" — its center of mass, assuming uniform density.

Bound an unfamiliar integrand above or below by a known one to determine convergence without evaluating.

Rewriting x^2+bx+c as (x+h)^2+k so a trig or u-substitution can apply.

The +C appended to an indefinite integral, capturing every antiderivative at once.

x=r\cos\theta,\ y=r\sin\theta one way; r^2=x^2+y^2,\ \tan\theta=y/x the other.

The path traced by a point on a rolling circle's rim: x=r(t-\sin t), y=r(1-\cos t).

The exact signed area under a curve, \int_a^b f(x)\,dx — the limit of a Riemann sum as n\to\infty.

An equation relating a function to its own derivative(s), like y'=ky.

Bound an unfamiliar series above or below by a known one to determine convergence without a formula.

Volume of revolution as stacked circular disks: V=\pi\int_a^b\left[r(x)\right]^2dx.

Net displacement is the signed integral of velocity; total distance integrates |v(t)| instead.

If \lim_{n\to\infty}a_n\ne0 (or doesn't exist), \sum a_n diverges — but a_n\to0 proves NOTHING about convergence.

Solving one parametric equation for t and substituting into the other to get a single x-y equation.

e^{i\theta}=\cos\theta+i\sin\theta — unifying the exponential and trig Maclaurin series via a complex exponent.

Numerically approximating a solution curve by taking small straight-line steps, y_{n+1}=y_n+h\cdot f(x_n,y_n).

On a symmetric interval [-a,a], an even integrand doubles; an odd integrand vanishes.

E[X]=\int x\,f(x)\,dx — the long-run average value X takes, weighted by its density.

y(t)=y_0e^{kt} — the solution to y'=ky, growing if k>0, decaying if k<0.

Differentiating an accumulation function g(x)=\int_a^x f(t)\,dt just returns f(x).

\int_a^b f(x)\,dx=F(b)-F(a) for any antiderivative F of f — the evaluation shortcut.

\sum ar^{n} — converges to \dfrac{a}{1-r} exactly when |r|<1, diverges otherwise.

\sum\dfrac1n=1+\dfrac12+\dfrac13+\cdots — diverges, despite terms shrinking to 0.

F=\int(\text{weight density})\times(\text{depth})\times(\text{width})\,d(\text{depth}) on a submerged surface.

A definite integral with an infinite bound, or an integrand that blows up somewhere in the interval.

The notation \int f(x)\,dx for the entire family of antiderivatives of f.

A limit expression like \frac{0}{0} or \frac{\infty}{\infty} whose value isn't determined by the form alone.

A series \sum a_n is the limit of its partial sums S_N=\sum_{n=1}^N a_n as N\to\infty.

A differential equation plus a known starting point, together pinning down ONE specific solution.

\sum a_n and \int_1^{\infty}f(x)\,dx converge or diverge TOGETHER, when a_n=f(n) for positive, decreasing, continuous f.

A multiplier \mu(x)=e^{\int P(x)\,dx} that turns a linear DE's left side into a product-rule derivative.

\int u\,dv=uv-\int v\,du — reversing the product rule for a product integrand.

For \frac00 or \frac{\infty}{\infty} forms, \lim\frac{f(x)}{g(x)}=\lim\frac{f'(x)}{g'(x)}.

R_n(x)=f(x)-P_n(x), bounded by \dfrac{M}{(n+1)!}|x-a|^{n+1} — how far a Taylor polynomial can be from f.

The three standard choices of sample point within each Riemann-sum rectangle.

If \lim\dfrac{a_n}{b_n} is a finite positive number, \sum a_n and \sum b_n converge or diverge TOGETHER.

The value a_n approaches as n\to\infty; a sequence CONVERGES if this limit exists, DIVERGES otherwise.

y'=ky\left(1-\dfrac{y}{M}\right) — growth proportional to size AND remaining room, capped at carrying capacity M.

The handful of standard Maclaurin series (e^x, \sin x, \cos x, \ln(1+x), \frac{1}{1-x}, ...) worth memorizing outright.

A continuous function actually ATTAINS its average value somewhere on the interval.

A tank differential equation tracking dissolved substance as fluid flows in and out.

A sequence that's always increasing (or always decreasing) AND stays within some fixed range — guaranteed to converge.

\int_a^b f'(x)\,dx=f(b)-f(a) — a rate's integral is the total change in the quantity itself.

The area a definite integral computes — area above the axis counts positive, below counts negative.

T(t)=T_{\text{env}}+(T_0-T_{\text{env}})e^{-kt} — temperature decays exponentially TOWARD the surroundings, not to zero.

\sum\dfrac{1}{n^p} — converges exactly when p>1, diverges otherwise (including the harmonic series at p=1).

L=\int_a^b\sqrt{[x'(t)]^2+[y'(t)]^2}\,dt — the total distance traveled along a parametric curve.

A curve traced by x(t) and y(t) moving together as a third variable t (often time) increases.

\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt} — the slope of a parametric curve, via the chain rule.

\sqrt{[x'(t)]^2+[y'(t)]^2} — the rate of distance traveled along the curve, the arc-length integrand.

Splitting a rational function into a sum of simpler fractions, each easy to integrate on its own.

L=\int_{\alpha}^{\beta}\sqrt{[r(\theta)]^2+[r'(\theta)]^2}\,d\theta — total distance traced along a polar curve.

A=\dfrac12\int_{\alpha}^{\beta}\left[r(\theta)\right]^2d\theta — area swept out by a polar curve, via infinitesimal sectors.

A point located by distance r from the origin and angle \theta from the positive x-axis, instead of (x,y).

Standard named shapes traced by simple polar equations: r=a\cos(n\theta), r=a\pm a\cos\theta, r=a\pm b\cos\theta.

Points shared by two polar curves — trickier than rectangular intersections, since the SAME point has many (r,\theta) names.

\dfrac{dy}{dx}=\dfrac{dr/d\theta\cdot\sin\theta+r\cos\theta}{dr/d\theta\cdot\cos\theta-r\sin\theta} — via x=r\cos\theta,\ y=r\sin\theta treated parametrically.

\sum c_n(x-a)^n — an infinite series that's also a function of x, generalizing a polynomial to infinitely many terms.

A function f(x)\ge0 with \int f(x)\,dx=1 over its domain, whose integral over an interval gives a probability.

A power series converges within a distance R of its center; the endpoints need separate checking.

L=\lim\left|\dfrac{a_{n+1}}{a_n}\right|: L<1 converges absolutely, L>1 diverges, L=1 is inconclusive.

The Weierstrass substitution t=\tan(x/2), turning any rational function of \sin x,\cos x into a rational function of t.

A sequence defined from its own previous term(s); IF it converges, its limit must satisfy L=f(L).

An approximation of the area under a curve, built from a sum of rectangle areas.

L=\lim\sqrt[n]{|a_n|}: L<1 converges absolutely, L>1 diverges, L=1 is inconclusive.

One where the x's and y's can be sorted onto opposite sides, then each side integrated separately.

An ordered, infinite list of numbers a_1,a_2,a_3,\ldots — really just a function of the index n.

Integrating a Taylor series term by term to handle an integrand like e^{-x^2} with no elementary antiderivative.

Replacing each piece of a limit expression with its Taylor series to cancel indeterminate behavior algebraically.

Volume of revolution as nested cylindrical shells: V=2\pi\int_a^b x\cdot h(x)\,dx.

Approximating a definite integral by fitting a parabola through each pair of adjacent subintervals.

A grid of short line segments, each showing dy/dx at that point — a visual map of a differential equation's solutions.

S=\int_a^b 2\pi y\sqrt{1+[y']^2}\,dx — the area of the surface traced by revolving a curve.

A shortcut for integration by parts applied repeatedly, via a two-column derivative/integral table.

The degree-n polynomial matching f and its first n derivatives at x=a — the best local polynomial approximation.

The infinite power series \sum\dfrac{f^{(n)}(a)}{n!}(x-a)^n built from ALL of f's derivatives; a Maclaurin series is the special case a=0.

Replacing a nonlinear function with just its degree-1 Taylor polynomial for a good LOCAL approximation.

A series whose partial sums collapse via cancellation, leaving only a few surviving terms.

Approximating a definite integral by summing trapezoids instead of rectangles.

Substituting x=a\sin\theta, a\tan\theta, or a\sec\theta to eliminate a square root of a quadratic.

Reversing the chain rule to integrate a composite function, by renaming its inner piece u.

Volume of revolution with a hole: V=\pi\int_a^b\left(\left[R(x)\right]^2-\left[r(x)\right]^2\right)dx.

W=\int_a^b F(x)\,dx — total work done by a variable force, summed over displacement.