Triple Integrals

Concepts/Skills (May or may not require answering in questions)

100

A Riemann sum for solids chopped into blocks (guess a question)

What is a triple integral?

200

Determine if possible to solve or not; if possible, solve: ∭dV=∫∫∫dxdydz, under plane x-y+z=1 where inner limits: [-y-z, y+z], middle limits: [1-z,1], outer limits: [2,3] (do not guess a question; answer with a number or label as counterexample)

∫∫∫dV=∫∫∫dxdydz=∫∫xdydz=∫∫(2y+2z)dydz=∫∫(y^2)dz= ∫(1^2-(1-z)^2)dz=(1/3)(1-z)^3=(-8/3)-(-1/3)=-7/3

300

Determine if possible to solve or not; if possible, solve: ∫∫∫dV, dV=dxdydz, outer limits=[y-z,1], middle limits=[z,1], inner limits= [0,1]

Can't do :P counterexample; outer integral (x) defined by y and z, but y and z not even defined yet

Vocabulary/Notation

100

∫∫∫

What is the symbol used with integrals?

200

A triple integral rewritten as three single integrals

What is an iterated triple integral?

300

Ex. triple integral in the form of: ∑ijkf(xijk,yijk,zijk)ΔV (what is this written as?)

What is a Riemann sum?

Methods

100

Shadow Method; Cross Method (what are they used for?)

What are used to find the limits of integration?

200

Method used to rewrite triple integrals as a double integral within a single integral; chops a three dimensional region into slices

What is the cross section method?

300

Method used to rewrite a triple integral as a single integral within a double integral; imagine the sun is above a vertical axis

What is the shadow method?

Rules for Triple Integrals

100

These limits must be constant; cannot depend on other variables

What is the rule about outer limits?

200

These limits can depend on constants or one variable.

What is the rule for the middle limits?

300

These limits can depend on constants, one or two variables.

What is the rule for inner limits?

Problems(answer in actual numbers, not questions)

100

∫∫∫xdV, x:[0,1], y: [0,1], z: [0,1], dV=dxdydz

∫∫∫xdV=∫∫∫xdxdydz=∫∫(1/2)x^2dydz=∫∫(1/2)dydz=∫(1/2)dz= 1/2

200

The volume of a gas is (∫∫∫dV) L, where dV=dzdydx; find the volume given the limits of x: [0,1], y: [0,1], z: [0,1]. Watch units!

∫∫∫dV=∫∫∫dzdydx=∫∫dydx=∫dx=1 L

300

Find the mass of a gas with volume (∫∫∫dV) L, where dV=dxdydz; find the volume given the limits of x: [0,1], y: [0,1], z: [0,2]. The density p of the gas is modeled by (xyz) kilograms per L . Watch units! Hint: density=mass*volume

∫∫∫pdV=∫∫∫xyzdxdydz=∫∫(1/2)(x^2)yzdydz=(1/2)∫∫yzdydz= (1/2)∫(1/2)(y^2)zdz=(1/4)∫zdz= (1/4)(1/2)(z^2) =1/2 kg