Multiple Integration

Question 1

Volume between two equal regions between 3d curvatures

Answer 1

Volume is equal to the double integral of the multivariable function

V=∫∫(g(x,y)-f(x,y))dA

Question 2

Average value of a 3d surface function within a given region

Answer 2

(∫∫f(x,y)dA)/(RegionArea)

Question 3

Volume for a non rectangular regions

Answer 3

∫∫ f(x,y)

set second interval from one function value to another

Question 4

Volumes of additional regions

Answer 4

Always the sum
∫∫total(x)=∫∫f1(x)+∫∫f2(x)
Applies for triple integrals too

Question 5

Area within a polar curve section

Answer 5

∫∫f(r,θ)=∫∫f(r,θ) r dr dθ

First interval- largest angle, smallest angle
Second Interval- larger radius, smaller radius

Question 6

Area between two polar functions

Answer 6

∫∫f(r,θ)=∫∫f(r,θ) r dr dθ

First interval- largest angle, smallest angle
Second Interval- greater function, smaller function

Question 7

Volume under a function of three variables

Answer 7

∫∫∫f(x,y,z)dV

First Interval- b,a (region within the x-coordinate)
Second Interval- upper function on xy-axis, lower curve on xy-axis (describes the region)
Third Interval- upper function on 3d-axis, lower surface on 3d-axis (describes the surfaces)

Question 8

Average value of a three variable function

Answer 8

=(∫∫∫f(x,y,z)dV)/(Total Volume)

Question 9

Volume of Cylindrical Coordinates

Answer 9

∫∫∫f(r,θ,z)

First integral- from greater angle from x, to lesser angle from x
Second integral- greater area function given θ, lesser area function given θ
Third integral- Greater height function of (rcosθ,rsinθ), and the lesser height function

Question 10

Ranges for cylindrical coordinates given volume

Answer 10

∫∫∫f(r,θ,z)

θ- Within the first interval
r- Within the second interval
z- Within the third interval

Question 11

Volume of a spherical portion

Answer 11

∫∫∫f(p,ϕ,θ)p^2(sinϕ)dp,dϕ,dθ

First integral- from greater angle from x, to lesser angle from x
Second integral- greater angle from z-coordinate, lesser angle from z-coordinate
Third integral- larger sphere, smaller sphere

Question 12

Coordinate ranges within spherical volumes

Answer 12

∫∫∫f(R,ϕ,θ)R^2(sinϕ)dR,dϕ,dθ

θ- within first integral
ϕ- within second integral
R- radius of larger sphere, radius smaller sphere

Question 13

Center of mass for complex objects in a 3d coordinate system

Answer 13

X-coordinate from zero position=(∫∫∫xp(x,y,z)/(∫∫∫p(x,y,z)
Y-coordinate from zero position=(∫∫∫yp(x,y,z)/(∫∫∫p(x,y,z)
Z-coordinate from zero position=(∫∫∫z*p(x,y,z)/(∫∫∫p(x,y,z)

Density functions, and the product of their linear position

Question 14

Xy-coordinate center of mass for flat objects

Answer 14

X-coordinate from zero position=(∫∫xp(x,y)/(∫∫p(x,y)
Y-coordinate from zero position=(∫∫yp(x,y)/(∫∫p(x,y)

Density functions, and the product of their linear position

Question 15

Change of variables in multiple integration

Answer 15

Volume remains the same, but the region, interval, and surface functions change with time

Question 16

Jacobean Determinant (of two variable transformation)

Answer 16

Matrix determinant of a two-by-two matrix of partial derivatives of the change in variables in multiple integration
x=g(u,v) y=h(u,v)
J(u,v)=∂(x,y)/∂(u,v)=[(∂x/∂u)(∂y/∂v)-(∂y/∂u)(∂x/∂v)]

Question 17

Change of variables formula for double integrals

Answer 17

∫∫f(x,y)=∫∫f(g(u,v),h(u,v))|J(u,v)|

Question 18

Jacobean Determinant (of three variable transformation)

Answer 18

Matrix determinant of a three-by-three matrix of partial derivatives of the change in variables in multiple integration
x=g(u,v,w) y=h(u,v,w)
J(u,v,w)=∂(x,y,z)/∂(u,v,w)

Question 19

Change of variables formula for triple integration

Answer 19

∫∫∫f(x,y,z)=∫∫∫f(g(u,v,w),h(u,v,w),p(u,v,w)|J(u,v,w)|

Question 20

The goal of changing variables in integration problems

Answer 20

Integrand Simplicity

Question 21

Selecting new variables (functions) based on simplicity

Answer 21

1) Solve the integration for volume or area so that you know how much you have to work with here
2) Identify the simplest variables, constants on the xyz coordinate plane. (Now the uvp coordinates)
3) Write a simple function of (x,y,z) to produce (u,v,p)
4) Look at the integrand to determine functions for uvp to simplify the statement
5) Look at the graph to determine more simply geometry of area or volume
6) Intuition- You need to practice this!!!

Question 22

Net area under a curve drawn in 3d space

Answer 22

∫f(x(t),y(t),z(t))√(x’(t)^2+y’(t)^2+z’(t)^2)dt

Integral of the quantity (3d curve times magnitude of the derivative)

Question 23

Volume under 3d curvature

Answer 23

Volume is equal to the double integral of the multivariable function
V=∫∫f(x,y)dA