The angle at the vertex is 2/3, and the top is flat and at a height of 2/3. Writ
ID: 2884231 • Letter: T
Question
The angle at the vertex is 2/3, and the top is flat and at a height of 2/3. Write the limits of integration for WdV in the following coordinates (do not reduce the domain of integration by taking advantage of symmetry)
(a) Cartesian:
With a= , b= ,
c= , d= ,
e= , and f= ,
Volume = badcfe d d d
(b) Cylindrical:
With a= , b= ,
c= , d= ,
e= , and f= ,
Volume = badcfe d d d
(c) Spherical:
With a= , b= ,
c= , d= ,
e= , and f= ,
Volume = badcfe d d d
Explanation / Answer
a) cartesian :
tan=(x2+y2)/z
=(1/2)*(2/3)=(/3)
tan(/3)=(x2+y2)/z
3=(x2+y2)/z
z=((x2+y2)/3)
((x2+y2)/3) <= z<=2/3
((x2+y2)/3) =2/3
(x2+y2)/3 =4/3
(x2+y2)=4
-(4-x2)<=y<=(4-x2) ,-2<=x<=2
a =-2,b=2,c=-(4-x2),d=(4-x2),e=((x2+y2)/3) ,f=2/3
volume=[-2 to 2][-(4-x2) to (4-x2)][((x2+y2)/3) to 2/3] dz dy dx
-----------------------------------------------
cylindrical:
in cylindrical coordinates
x=rcos, y=rsin
x2+y2=r2
x2+y2=4=22
((x2+y2)/3) <= z<=2/3
r/3 <= z<=2/3
0<=<=2,0<=r<=2 ,r/3 <= z<=2/3
dv =r dz dr d
a=0,b=2,c=0,d=2,e=r/3,f=2/3
volume=[0 to 2] [0 to 2] [r/3 to 2/3] r dz dr d
------------------------------------------------
spherical:
in spherical coordinates
x=sincos,y=sinsin,z=cos
x2+y2+z2=2
z=2/3
cos=2/3
=(2/3)sec
==>0<=<=(2/3)sec
dv=2sin d d d
0<=<=2,0<=<=/3,0<=<=(2/3)sec
a=0,b=2,c=0,d=/3,e=0,f=(2/3)sec
volume= [0 to 2] [0 to /3] [0 to (2/3)sec] 2sin d d d
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