Key Equations

Key Equations

Area between two curves, integrating on the x-axis	$A={{\displaystyle \int }}_a^b\left[f(x)-g(x)\right]dx$
Area between two curves, integrating on the y-axis	$A={{\displaystyle \int }}_c^d\left[u(y)-v(y)\right]dy$

Disk Method along the x-axis	$V={\displaystyle {\int }_a^b\pi {\left[f\left(x\right)\right]}^{2}dx}$
Disk Method along the y-axis	$V={\displaystyle {\int }_c^d\pi {\left[g\left(y\right)\right]}^{2}dy}$
Washer Method	$V={\displaystyle {\int }_a^b\pi \left[{\left(f\left(x\right)\right)}^2-{\left(g\left(x\right)\right)}^2\right]}dx$

Method of Cylindrical Shells

$V={\displaystyle {\int }_a^b\left(2\pi xf(x)\right)}dx$

Arc Length of a Function of x	$\text{Arc Length}={\displaystyle {\int }_a^b\sqrt{1+{\left[f^{\prime }(x)\right]}^2}}dx$
Arc Length of a Function of y	$\text{Arc Length}={\displaystyle {\int }_c^d\sqrt{1+{\left[g^{\prime }(y)\right]}^2}}dy$
Surface Area of a Function of x	$\text{Surface Area}={\displaystyle {\int }_a^b\left(2\pi f(x)\sqrt{1+{\left(f^{\prime }(x)\right)}^2}\right)dx}$

Mass of a one-dimensional object	$m={\displaystyle {\int }_a^b\rho }(x)dx$
Mass of a circular object	$m={\displaystyle {\int }_0^{r}2}\pi x\rho (x)dx$
Work done on an object	$W={\displaystyle {\int }_a^{b}F}(x)dx$
Hydrostatic force on a plate	$F={\displaystyle {\int }_a^b\rho }w(x)s(x)dx$

Mass of a lamina	$m=\rho {{\displaystyle \int }}_a^{b}f(x)dx$
Moments of a lamina	$M_x=\rho {\displaystyle {\int }_a^b\frac{{\left[f(x)\right]}^2}{2}dx}\text{and}{M}_y=\rho {\displaystyle {\int }_a^{b}xf(x)dx}$
Center of mass of a lamina	$\bar{x}=\frac{M_y}{m}\text{and}\bar{y}=\frac{M_x}{m}$

Natural logarithm function	$\text{ln}x={\displaystyle {\int }_1^x\frac{1}{t}}dt$ Z
Exponential function $y=e^x$	$\text{ln}y=\text{ln}\left(e^x\right)=x$ Z