Definite integrals {improper integral} can have infinity as limit or integrate over an open interval. To integrate improper integrals, take integral limit as variable approaches infinity. If limit is infinity, split domain at value zero, integrate over both intervals, and add results.
To find curved-figure areas and volumes, add many small discrete triangular or trapezoidal areas {method of exhaustion} {exhaustion method}.
Magnitudes can be infinite numbers of small units {indivisibles method} {method of indivisibles}. Cavalieri invented this calculus forerunner [1629].
For degree-n functions, definite integral over (a,b) is ((b - a) / (3*n)) * (sum from k = 0 to k = n of c * f(a + k * (b - a) / n)), where n is even integer, c = 4 if k is odd, c = 2 if k is even, and c = 1 if k = 0 {Simpson's rule} {Simpson rule}. Error is less than or equal to M * (b - a)^5 / (180 * n^4), where M is less than or equal to fourth-derivative absolute value.
For functions sin^m(x), cos^m(x), and cos^m(x) * sin^m(x), where m is positive integer, such as sin(x) and sin^2(x), reduction formulas {Wallis's formula} {Wallis formula} can evaluate definite integrals from x = 0 to x = pi/2.
Integral of u*dv equals u*v minus integral of v*du {parts integration} {integration by parts}.
For two functions that depend on the same variable {composite function integration}, integral of u(x) * dv(x) = u(x) * v(x) - integral of v(x) * du(x).
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Date Modified: 2022.0225