Mathematical formulas

Calculus II

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Sections

1Notation, domains and conventions
2Indefinite integral and properties
3Definite integral and Fundamental Theorem of Calculus
4Substitution method
5Integration by parts
6Trigonometric integrals
7Trigonometric substitution and quadratic radicals
8Rational functions and partial fractions
9Improper integrals
10Applications of the integral
11Numerical integration
12Parametric curves
13Polar coordinates
14Sequences and series
15Power series and Taylor series
16Elementary differential equations
17Guide to choosing a method
AAppendix A: extensive table of antiderivatives
BAppendix B: useful equivalences

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Subjects
Calculus II
Mass, moments, and centroid of a lamina
For a region between $y = f (x)$ and $y = g (x)$ , with constant surface density $ρ$ :

10.9. Mass, moments, and centroid of a lamina

For a region between $y = f (x)$ and $y = g (x)$ , with constant surface density $m = ρ \int_{a}^{b} [f (x) - g (x)] d x$ :

In 10.9. Mass, moments, and centroid of a lamina

Formula

m = ρ \int_{a}^{b} [f (x) - g (x)] d x

Variables

$x$ Variable independiente (variable de integración)
$a$ Constant real (a menudo $a > 0$ )
$b$ Constant real
$m$ Exponente o índice entero
$f$ Función integrando
$g$ Función auxiliar (a menudo en composición o dominio)
$d x$ Diferencial respecto de $x$

Related formulas

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Formula
$M_{y} = ρ \int_{a}^{b} x [f (x) - g (x)] d x$
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Formula
$M_{x} = \frac{ρ}{2} \int_{a}^{b} [f (x)^{2} - g (x)^{2}] d x$
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Formula
$\overset{x}{ˉ} = \frac{M _{y}}{m}, \overset{y}{ˉ} = \frac{M _{x}}{m} .$
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