Triple Integrals

Solving

Conditions

• $E$ is closed and bounded by a smooth surface
• $f$ is continuous on $E$

When Simple in a Direction

When E is z-simple (could work along any axis, just need to swap the variables):

$${\iiint }_{E}f(x,y,z)dV={\iint }_{D_{xy}}{\int }_{z_b(x,y)}^{z_t(x,y)}f(x,y,z)dzdA$$

Triple Fubini

$$\begin{gathered} E=[a,b]\times [c,d]\times [s,t] \\ {\iiint }_{E}f(x,y,z)dv={\int }_a^b{\int }_c^d{\int }_s^{t}f(x,y,z)dzdydx \end{gathered}$$

Change of Variables

• Cylindrical Change of Variables
• Spherical Change of Variables
• General Triple Change of Variables