13.7 Cylindrical Coordinates

# 13.7 Cylindrical Coordinates - 1 3.7  C y l in d r ic a l...

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Unformatted text preview: 1 3 .7  C y l in d r ic a l  C o o r d in a t e s   C y l in d r ic a l   z H r, q , zL y x   r  ≥  0 ,  r  is  t h e  d is t a n c e  fr o m  t h e  o r ig in  t o  t h e  p r o j e c t io n   P ′  in  t h e  x y  p l a n e .  0 ≤ θ ≤ 2π , θ  is the angle from the x‐axis to the line  b e t w e e n  t h e  o r ig in  a n d  p r o je c t io n  o f t h e  p o in t  ( P ′ )  in   t h e  x y  p l a n e .    C o n v e r s i o n   e q u a t i o n s :  x = r cosθ y tan θ = x y = r sin θ z=z x 2 + y 2 = r2   E x .  G r a p h  r  =  1               E x .  G r a p h   θ = π 4              E x .  G r a p h  z  =  ‐ 2           E x . C o n v e r t  t h e  e q u a t io n  6 x  =  x 2 +  y 2 t o  c y l in d r ic a l   c o o r d in a t e s .                    3− z r= E x .  C o n v e r t   2 sin θ  to Cartesian coordinates.                    ∫∫∫ f (r,θ, z) E dV = ∫∫∫ f ( r,θ, z) _________ E     Ex. Convert the following integral to cylindrical coordinates    1 ∫∫ 1− x 2 −1 − 1− x                         2 ∫ 2− x 2 − y 2 x +y 2 2 (x 2 +y 3 22 ) dzdydx   Ex.  Suppose the solid looks like         Ex.  Find the volume of the solid bounded by x2 + y2 = z2 and   x2 + y2 = 4.                            ∫∫∫ x Ex.  Set up integrals to evaluate  E dV  where E is the  solid bounded by x2 + y2 = z and z = 18 – x2 – y2.                                                        Ex.  Set up integrals to find the center of mass of the solid  bounded by x2 + y2 = 2y,  z = y + 1, and z = 0.                                                            Do: Find the volume of the solid inside the sphere x2 + y2 + z2 =  2 and outside the cylinder x2 + y2 = 1 using cylindrical  coordinates.                ...
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