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Calc 3 12.6-12.7

 

<== Calculus 3
Number Question Answer
See the work
1. What are the rectangular coordinates of the point whose cylindrical coordinates are
(r= 4 ,\  	heta = rac{2 \pi}{5}  ,\ z= 1 ) ?


x=
y=
z=

x= 1.236
y= 3.804
z= 1
The work
2. What are the rectangular coordinates of the point whose cylindrical coordinates
are (r = 5, &theta = 1.4, z = 6 ) ?
x =
y =
z =
x= 0.8498
y= 4.927
z= 6
The work
3. What are the cylindrical coordinates of the point whose rectangular coordinates are (x = 5, y = 1 , z = -2 ) ?


r =
&theta =
z =

r = 5.099
&theta = 0.1974
z = -2
The work
4. What are the cylindrical coordinates of the point whose rectangular coordinates are (x = -3 , y = 2 , z = 1 ) ?


r =
&theta =
z =

r = 3.6056
&theta = 2.554
z = 1
The work
5. Express the point given in Cartesian coordinates in cylindrical coordinates (r, &theta, z).

A) (7 \sqrt{3}/2, 7 (1/2), 7) = (,, )
B)(-7 \sqrt{3}/2, 7 (1/2), 7) = (,, )
C)(7 \sqrt{3}/2, -7 (1/2), 7) = (,, )
D)(-7 \sqrt{3}/2, -7 (1/2), 7) = (,, )

A) ( 7, 0.5236, 7)
B) ( 7, 2.61799, 7)
C) ( 7, 5.75959, 7)
D) ( 7, 3.66519, 7)
The work
6. Use cylindrical coordinates to evaluate the triple integral \iiint_E \, \sqrt{x^{2} + y^{2}} \, dV, where E is the solid bounded by the circular paraboloid z = 16 - \left( x^{2} + y^{2} 
ight) and the xy-plane.

857.864 The work
7. Find the volume of the solid enclosed by the paraboloids z =
4 \left( x^{2} + y^{2} 
ight) and z = 8 -
4 \left( x^{2} + y^{2} 
ight).

12.5664 The work
8. Find the volume of the ellipsoid x^2 + y^2 + 8 z^2 = 36.

319.89 The work
9. What are the rectangular coordinates of the point whose spherical coordinates are
(1 , 3&pi/6, 4&pi/6 ) ?


x =
y =
z =

x = 0
y = 0.866
z = -0.5
The work
10. What are the spherical coordinates of the point whose rectangular coordinates are
(4, 4,4) ?


&rho =
&theta =
&phi =

&rho = 6.9282
&theta = 0.7854
&phi = 0.95532
The work
11. Express the point given in Cartesian coordinates in spherical coordinates (&rho, &theta, &phi).

A) (7 (1/2) (\sqrt{2}/2), 7 (1/2) (\sqrt{2}/2) , 7 \sqrt{3}/2) = (,,)
B)(-7 (1/2) (\sqrt{2}/2), 7 (1/2) (\sqrt{2}/2) , 7 \sqrt{3}/2) = (,, )
C)(7 (1/2) (\sqrt{2}/2), -7 (1/2) (\sqrt{2}/2) , -7 \sqrt{3}/2) = (,, )
D)(-7 (1/2) (\sqrt{2}/2), -7 (1/2) (\sqrt{2}/2) , -7 \sqrt{3}/2) = (,, )

A) ( 7, 0.7854, 0.5234)
B) ( 7, 2.3562, 0.5236)
C) ( 7, 5.4978, 2.618)
D) ( 7, 3.927, 2.618)
The work
12. What are the cylindrical coordinates of the point whose spherical coordinates are
( 5, -5, &pi/6) ?


r =
&theta =
z =

r = 2.5
&theta = -5
z = 4.3301
The work
13. Use spherical coordinates to evaluate the triple integral \iiint_E \, (x^{2} + y^{2} + z^{2}) \, dV, where E is the ball: x^{2} + y^{2} + z^{2} \leq 81.

148406.324 The work
14. Use spherical coordinates to evaluate the triple integral \displaystyle \iiint_E \, rac{e^{-(x^{2} + y^{2} +
z^{2})}}{\sqrt{x^{2} + y^{2} + z^{2}}} \, dV, where E is the region bounded by the spheres x^{2} + y^{2} + z^{2} = 49 and x^{2} + y^{2} + z^{2} = 100.

0 The work
15. FInd the volume of the solid that lies within the sphere x^2 + y^2 +  z^2 = 9, above the xy plane, and outside the cone z = 6 \sqrt{x^2+y^2}.

55.779 The work
16. question Answer The work
17. question Answer The work