Key Equations - Calculus Volume 2

Area between two curves, integrating on the x-axis	$A = \int_{a}^{b} [f (x) - g (x)] d x$
Area between two curves, integrating on the y-axis	$A = \int_{c}^{d} [u (y) - v (y)] d y$

Method of Cylindrical Shells

V = \int_{a}^{b} (2 π x f (x)) d x

Arc Length of a Function of x	$Arc Length = \int_{a}^{b} \sqrt{1 + {[f^{'} (x)]}^{2}} d x$
Arc Length of a Function of y	$Arc Length = \int_{c}^{d} \sqrt{1 + {[g^{'} (y)]}^{2}} d y$
Surface Area of a Function of x	$Surface Area = \int_{a}^{b} (2 π f (x) \sqrt{1 + {(f^{'} (x))}^{2}}) d x$

Mass of a one-dimensional object	$m = \int_{a}^{b} ρ (x) d x$
Mass of a circular object	$m = \int_{0}^{r} 2 π x ρ (x) d x$
Work done on an object	$W = \int_{a}^{b} F (x) d x$
Hydrostatic force on a plate	$F = \int_{a}^{b} ρ w (x) s (x) d x$

Mass of a lamina	$m = ρ \int_{a}^{b} f (x) d x$
Moments of a lamina	$M_{x} = ρ \int_{a}^{b} \frac{{[f (x)]}^{2}}{2} d x and M_{y} = ρ \int_{a}^{b} x f (x) d x$
Center of mass of a lamina	$\bar{x} = \frac{M_{y}}{m} and \bar{y} = \frac{M_{x}}{m}$

Natural logarithm function	$ln x = \int_{1}^{x} \frac{1}{t} d t$ Z
Exponential function $y = e^{x}$	$ln y = ln (e^{x}) = x$ Z