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Multiple integral (Tripple integral (Cylindrical coordinate (∫∫∫f(r,θ…
Multiple integral
Tripple integral
Rectangular coordinate
∫∫∫f(x,y,z)dV
∫∫∫f(x,y,z)dxdA
∫∫∫f(x,y,z)dxdydz
∫∫∫f(x,y,z)dxdzdy
∫∫∫f(x,y,z)dydV
∫∫∫f(x,y,z)dydxdz
∫∫∫f(x,y,z)dydzdx
∫∫∫f(x,y,z)dzdV
∫∫∫f(x,y,z)dzdxdy
∫∫∫f(x,y,z)dzdydx
Cylindrical coordinate
∫∫∫f(r,θ,z)rdzdrdθ
∫∫∫f(r,y,θ)rdydrdθ
∫∫∫f(x,r,θ)rdxdrdθ
Spherical cooradinate
necessary formula
r = ρsinϕ
z = ρcosϕ
∫∫∫f(ρ,ϕ,θ)ρ^2sinϕdρdϕdθ
Application
f(x,y,z) = 1
Find the volume
Double integral
Rectangular coordinate
∫∫f(x,y)dA
∫∫f(x,y)dydx
∫∫f(x,y)dxdy
Reverse dxdy <-> dydx
Categories
Rectangular
Non-rectangular
Cylindrical coordinate
∫∫f(r,θ)rdrdθ
necessary formular
x = rcose
y = rsin
r^2 = x^2 + y^2
Application
f(x,y) = 1
Find the area of region R
f(x,y) = z
Find the volume
Polar coordinate
Conversion
rectangular -> polar
polar -> rectangular
Drawing graphs
r = a
r=2cosθ / r=2sinθ
r = a +- acosθ / r = a +- asinθ
r = a +- bcosθ / r = a +- bsinθ
a/b < 1
1 < a/b < 2
a/b > 2
