MAT 1236 Calculus III Section 15.7 Triple Integrals

HW WebAssign 15.7 (10 problems, 101 min.)

Preview Discuss the definitions of integrals of functions with 3 variables Discuss how the concept is the extension of the 1 and 2-variable case Computing triple integrals

1D Vs 2D Vs 3D

Intervals [a,b] Domains

(x,f(x)); Curves Graphs

Integrals

Triple Integral over a Box

Geometric Meanings

Fubini’s Theorem

Splitting Formula

Example 1

Special Regions

Example 2 Find the volume of E, where E is the solid bounded by

Expectations You are expected to put down E and D separately.

Special Regions

Before Next Example... We need to introduce a special type of surface. I added this an hour ago. So it is not on your HO. I usually drew all these pictures by hand. And it may be a good idea to use Maple to illustrate this. Try not to copy too much.

Paraboloids One of the most common surfaces They are related to parabolas Projections to the xz- and yz- planes are parabolas

Example 3 Find the volume of the solid bounded by the paraboloids

Example 3 Find the volume of the solid bounded by the paraboloids

Example 3 Find the volume of the solid bounded by the paraboloids

Example 3 Find the volume of the solid bounded by the paraboloids

Example 3 Find the volume of the solid bounded by the paraboloids

Go Nuts! GO NUTS!

Go Nuts! Please take one on your way out!