Triple Integration in Cylindrical Coordinates

University of Babylon Lecture: Hussein ALbermany
Collage of Material Engineering Subject: Mathematics
Department of Metallurgy Engineering Stage : 2nd stage
24
Triple Integration in Cylindrical Coordinates
To integrate continuous function f(r, ?, z) over a region given by :
z1(r,?)? z ? z2(r,?) ,
r1(?)? r ? r2(?),
?1 ? ? ? ?2
EXAMPLE 15: Find the centroid ( ? = 1 ) of the solid enclosed by the cylinder x2 + y2 = 4 bounded above by the paraboloid z=x2 + y2 , and bounded below by the xy-plane.
Solution:
x2 + y2 = 4 , (x2 + y2 = r2) r2 = 4 r= 2
z=x2 + y2 z = r2
z1(r,?)? z ? z2(r,?) : 0 ? z ? r2
r1(?)? r ? r2(?) : 0 ? r ? 2
?1 ? z ? ?2 : 0 ? ? ? 2??
University of Babylon Lecture: Hussein ALbermany
Collage of Material Engineering Subject: Mathematics
Department of Metallurgy Engineering Stage : 2nd stage
25
Spherical Coordinates and Integration
Spherical coordinates locate points in space with two angles and one distance
Spherical coordinates represent a point P in space by ordered triples in which
1. ? : Sphere radius
2. ? : angle between ? and positive z-axis (0 ? ? ? ?? ).
3. ? : angle between ? and x-axis (0 ? ? ? 2?? ).
To integrate continuous function f(r, ?, z) over a region given by :
?1(?,?) ? ? ? ?2(?,?) ,
?1(?) ? ? ? ?2(?),
?1 ? ? ? ?2
Coordinate Conversion Formulas
Volume:
University of Babylon Lecture: Hussein ALbermany
Collage of Material Engineering Subject: Mathematics
Department of Metallurgy Engineering Stage : 2nd stage
26
EXAMPLE 16: Find the volume of the “ice cream cone” D cut from the solid sphere ? ? 1 by the cone ?= ??/3. then find moment of inertia about the z-axis.
Solution:
0 ? ? ? 1 , 0 ? ? ? ??/3 , 0 ? ? ? 2??