Multiple Integration Flashcards
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
Average value of a 3d surface function within a given region
(∫∫f(x,y)dA)/(RegionArea)
Volume for a non rectangular regions
∫∫ f(x,y)
set second interval from one function value to another
Volumes of additional regions
Always the sum
∫∫total(x)=∫∫f1(x)+∫∫f2(x)
Applies for triple integrals too
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
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
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)
Average value of a three variable function
=(∫∫∫f(x,y,z)dV)/(Total Volume)
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 (rcosθ,rsinθ), and the lesser height function
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
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
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
Center of mass for complex objects in a 3d coordinate system
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
Xy-coordinate center of mass for flat objects
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
Change of variables in multiple integration
Volume remains the same, but the region, interval, and surface functions change with time