Appendix: Integral Fact Sheet

Single Variable Integrals

• $\large \int_a^b f(x)dx$ is the area under $\large f$ from $\large x=a$ to $\large x=b$.
• The average of the function $\large f$ over the $\large [a,b]$ interval is $\large \displaystyle \bar{f}=\frac{1}{b-a}\int_a^b f(x) dx$

Double Integrals

• $\large \iint_R f(x,y)dA$ is the volume under $\large f$ and above $\large R$.
• $\large \iint_R 1 dA$ is the area of the region $\large R$.
• The average value of a function $\large f(x,y)$ on the region $\large R$ is $\large \frac{1}{Area(R)}\iint_R f(x,y) dA$.
• If $\large \rho(x,y)$ is a mass density of a plate with shape $\large R$, then $\large \iint_R \rho(x,y) dA$ is a total mass.

Triple Integrals

• In general, $\large \iiint_D f(x,y,z) dV$ does not represent anything particular from a geometric point of view.
• $\large \iiint_D 1 dV$ is the volume of the region $\large D$.
• The average value of a function $\large f(x,y,z)$ on the region $\large D$ is $\large \frac{1}{Volume(D)}\iiint_D f(x,y,z) dV$.
• If $\large \rho(x,y,z)$ is a mass density of a solid with shape $\large D$, then $\large \iiint_D \rho(x,y,z) dV$ is a total mass.

Line Integrals

• $\large \int_{C}1 ds = \int_a^b|\vec{r'}(t)|dt$ is the length of the curve $\large C$ from $\large a$ to $\large b$ (called the arc length of $\large C$).
• The average value of the function $\large f$ on the curve $\large C$ is $\large \displaystyle \frac{\int_C f ds}{\int_C 1 ds}$ (the average is the integral divided by the length of the curve).
• If $\large f$ represents a density (charge density, mass density, etc.), then $\large \int_a^b f(x,y,z)ds$ is the total amount (of charge, mass, etc.) on the curve $\large C$.
• The line integral over a vector field $\large \int_C\vec{F}\cdot d\vec{s}$ represents the work done in moving an object through the force field $\large \vec{F}$ along the curve $\large C$.

Surface Integrals

• $\large \iint_S 1 dS$ is the surface area of $\large S$
• If $\large \rho(x,y,z)$ is a density on the surface, $\large \iint_S \rho(x,y,z) dS$ is the total amount (mass, charge, etc.).
• The average value of a function $\large f(x,y,z)$ on a surface $\large S$ is $\large \displaystyle \frac{\iint_S f(x,y,z) dS}{\iint_S 1 dS}$
• The surface integral over a vector field $\large \iint_S\vec{F}\cdot d\vec{S}$ represents the flux of the vector field through the surface $\large S$.