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Flashcards in Multiple Integration Deck (23):
0

Volume under 3d curvature

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

1

Volume between two equal regions between 3d curvatures

Volume is equal to the double integral of the multivariable function

V=∫∫(g(x,y)-f(x,y))dA

2

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

(∫∫f(x,y)dA)/(RegionArea)

3

Volume for a non rectangular regions

∫∫ f(x,y)
set second interval from one function value to another

4

Volumes of additional regions

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

5

Area within a polar curve section

∫∫f(r,θ)=∫∫f(r,θ) r dr dθ

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

6

Area between two polar functions

∫∫f(r,θ)=∫∫f(r,θ) r dr dθ

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

7

Volume under a function of three variables

∫∫∫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)

8

Average value of a three variable function

=(∫∫∫f(x,y,z)dV)/(Total Volume)

9

Volume of Cylindrical Coordinates

∫∫∫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 (r*cosθ,r*sinθ), and the lesser height function

10

Ranges for cylindrical coordinates given volume

∫∫∫f(r,θ,z)

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

11

Volume of a spherical portion

∫∫∫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

12

Coordinate ranges within spherical volumes

∫∫∫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

13

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

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

14

Xy-coordinate center of mass for flat objects

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

Density functions, and the product of their linear position

15

Change of variables in multiple integration

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

16

Jacobean Determinant (of two variable transformation)

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)]

17

Change of variables formula for double integrals

∫∫f(x,y)=∫∫f(g(u,v),h(u,v))|J(u,v)|

18

Jacobean Determinant (of three variable transformation)

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)

19

Change of variables formula for triple integration

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

20

The goal of changing variables in integration problems

Integrand Simplicity

21

Selecting new variables (functions) based on simplicity

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!!!

22

Net area under a curve drawn in 3d space

∫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)