Obtain the volume of the solid which is bounded by a circular paraboloid z=x^2+y^2, cylinder x^2+y^2=4, and Coordinate plane. And the solid is in the (x>=0, y>=0, z>=0).

ruigE 2020-11-27 Answered
Obtain the volume of the solid which is bounded by a circular paraboloid z=x2+y2, cylinder x2+y2=4, and Coordinate plane. And the solid is in the (x0,y0,z0).
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Expert Answer

Faiza Fuller
Answered 2020-11-28 Author has 108 answers
Cylindrical coordinates can be given as,
x=rcosθ
y=rsinθ
z=z
x2+y2=r2
Now the r will be 2 as x2+y2=22. Limits of region can be given as,
R{(r,θ,z):0r2,0θπ2,0zr2}
Volume of solid in cylindrical coordinate plane is given as,
V=Rrdrdθdz
Calculate the volume using the values,
V=0π2020r2rdzdrdθ
=0π202r[z]0r2drdθ
=0π202r3drdθ
=0π202[r44]02dθ
=0π24dθ
=4[te^]0π2
=4[π20]
=2π
Volume of the given solid is 2π
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