Applied Calculus: UNIT IV: Multiple Integrals
Triple Integrals: Cartesian coordinates into cylindrical polar coordinates
Explanation, Formula, Equation, Example and Solved Problems - Multiple Integrals: Triple Integrals: Cartesian coordinates into cylindrical polar coordinates
CARTESIAN
TO CYLINDRICAL POLAR COORDINATES
Let us first define
cylindrical coordinates of a point in space and derive the relation between
Cartesian and cylindrical coordinates for above figure
Let P be the point (x,
y, z) in Cartesian coordinate system. Let P M be drawn ⟂ r to the xoy plane and MN parallel to Oy. Let NOM = θ and OM = r. The triplet (r, θ, z) are called
cylindrical coordinate of P.
Clearly, ON = x = r cosθ;
NM = y = sinθ and MP = z
Thus the
transformations from three dimensional Cartesian to cylindrical coordinates are
x = r cosθ, y = r sinθ,
z= z.
Here x2 + y2 = r2
In this case,
Note:
Whenever ∫∫∫ f(x,y,z) dx dy dz is to
be evaluated throughout the volume of a right circular cylinder, it will be
advantageous to evaluate the corresponding triple integral in cylindrical
coordinates.
Example
80. By transforming into cylindrical coordinates, evaluate the integral ∫∫∫ (x2 + y2 + z2)
dx dy dz taken over the region of space defined by x2 + y2
≤ 1 and 0 ≤ z ≤ 1.
Solution:
The region of space is
the region enclosed by the cylinder x2
+ y2 = 1 whose base radius
is 1 and axis is the z‒axis and planes z = 0 and z = 1. The equation of the
cylinder in cylindrical coordinates is r = 1.
To convert from
Cartesian to cylindrical polar coordinates system, we have the following
transformation.
x = r cosθ, y = r sinθ,
z = z and dx dy dz= r dr dθ dz.
Also x2 + y2 = r2
x2
+ y2 ≤ 1
r2
≤ 1.
⇒
0 ≤ r ≤ 1.
In a cylinder, 0 ≤ θ ≤ 2π.
Given 0 ≤ z < 1.
Example
81. Evaluate 
by changing into cylindrical polar co‒ordinates.
Solution:
Here z = 0,z = 8
y = 0, y = √[ 4 − x2 ]
⇒
x2 + y2 = 4
⇒
r2 = 4 ⇒
r = 2
To change into
cylindrical co‒ordinates
x= rcosθ
y = r sinθ
z=z
Therefore the limits
are
r=0 ; r = 2
θ = 0 ; θ = π/2 (region
is in the 1st octant)
z = 0 ; z = 8 (given in
the integral)
= 512 / 3 cubic units
Applied Calculus: UNIT IV: Multiple Integrals : Tag: Applied Calculus : - Triple Integrals: Cartesian coordinates into cylindrical polar coordinates
Applied Calculus: UNIT IV: Multiple Integrals
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