Change from rectangular to cylindrical coordinates. (Let r\geq0 and 0\leq\theta\leq2\pi.) a) (-2, 2, 2) b) (-9,9\sqrt{3,6}) c) Use cylindrical coordin

coexpennan 2021-06-09 Answered
Change from rectangular to cylindrical coordinates. (Let r0 and 0θ2π.)
a) (2,2,2)
b) (9,93,6)
c) Use cylindrical coordinates.
Evaluate
ExdV
where E is enclosed by the planes z=0 and
z=x+y+10
and by the cylinders
x2+y2=16 and x2+y2=36
d) Use cylindrical coordinates.
Find the volume of the solid that is enclosed by the cone
z=x2+y2
and the sphere
x2+y2+z2=8.
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Expert Answer

wornoutwomanC
Answered 2021-06-10 Author has 81 answers
Step 1
Change from rectangular to cylindrical coordinates.
Let r0 and 0θ2π
a) (x,y,z)=(2,2,2)
Use cylindrical coordinates.
r=x2+y2=(2)3+23=22
θ=arctan(yx)=arctan(22)(1)=3π4
z=z=2
Therefore, the required coordinates is,
(r,θ,z)=(22,2π4,2)
Step 2
b) (x,y,z)=(9.93.6)
Use cylindrical coordinates.
r=x2+y2=(9)2+(93)2=18
θ=arctan(yx)=arctan(939)=arctan(3)=2π3
z=z=6
Therefore, the required coordinates is,
(r,θ,z)=(18,2π3,6)
Step 3
c) Use cylindrical coordinates, to evaluate ExdV
Where E is enclosed by the planes z=0 and z=x+y+10 and by the cylinders x2+y2=16 and x2+y2=36
16x2+y236
16r236 As x2+y2=r2
4r6
and 0θ2π
and 0zx+y+10
0zrcosθ+rsinθ+10 As x=rcosθ,y=rsinθ
and dV=dxdydz=rdrdθdz
So, ExdV=02a460rcosθ+rsinθ+10rcosθ(rdrdθdz)
=02π460rcosθ+rsinθ+10r2cosθdrdθdz
=02π46r2(rcosθ+rsinθ+10)cosθdrdθ
=02π46(r3(cos2θ+sinθcosθ)+10r2cosθ)drdθ

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