The surface area of revolution is the area of the surface made by spinning a curve around an axis. For y = f(x) from a to b spun around the x-axis, S = ∫ 2π f(x) √(1 + (f′(x))²) dx. Each tiny piece of the curve sweeps out a thin band whose area is about its circumference times its slanted width.
The arc length of a curve y = f(x) from x = a to x = b is L = ∫ √(1 + (f′(x))²) dx. The formula adds up tiny straight pieces of the curve, each found with the Pythagorean theorem. It works whenever f′ is continuous on [a, b].
The shell method finds the volume of a solid of revolution by slicing the region into thin strips parallel to the axis. Each strip spins into a cylindrical shell with volume about 2π · radius · height · thickness. Adding the shells gives V = ∫ 2π r(x) h(x) dx.
The comparison test for improper integrals decides convergence without integrating. If 0 ≤ g(x) ≤ f(x) and the integral of the bigger function f converges, then the integral of g converges too. If the integral of the smaller function g diverges, then the integral of f diverges too.
An improper integral with an infinite discontinuity has an integrand that blows up at an endpoint or inside the interval, like the integral of 1/√x from 0 to 1. Replace the bad point with t, integrate, and take a one-sided limit. A finite limit means the integral converges; otherwise it diverges.
An improper integral is a definite integral over an infinite interval, such as the integral of 1/x² from 1 to ∞. To find it, replace ∞ with a variable t, integrate, and take the limit as t → ∞. If the limit is a finite number, the integral converges to that number. Otherwise it diverges.
Simpson’s rule estimates a definite integral by fitting parabolas to the curve, two subintervals at a time. With an even number n of subintervals of width Δx, it adds the function values with weights 1, 4, 2, 4, …, 2, 4, 1 and multiplies by Δx/3. It is usually far more accurate than the trapezoidal rule.
To choose an integration technique, check the integral’s form in order: simplify it first, then try a basic rule, a u-substitution, integration by parts, a trig method or partial fractions. Each method fits a certain shape. Matching the shape to the method is what makes hard integrals manageable.
Hyperbolic functions are built from eˣ and e⁻ˣ: sinh x = (eˣ − e⁻ˣ)/2 and cosh x = (eˣ + e⁻ˣ)/2, with tanh x = sinh x / cosh x. They satisfy cosh²x − sinh²x = 1. Their derivatives are (sinh x)′ = cosh x and (cosh x)′ = sinh x, with no minus sign.
Integration by partial fractions integrates a rational function by splitting it into simpler fractions. Factor the denominator, write the fraction as a sum like A/(x − a) + B/(x − b), solve for the constants, and integrate each piece. Linear pieces give natural logs, and pieces over x² + a² can give arctangents.