Change y = x^2 to cylindrical coordinates. Change r = z/2 to rectangular coordin

Question

Change y = x^2 to cylindrical coordinates. Change r = z/2 to rectangular coordinates. Find the volume of the region consisting of the circular cylinder between the planes z = 0 and z = 5 - rcos theta. Be careful of the bounds of theta. Find the volume of the region between the cylinders x^2 + y^2 = 1 and x^2 + y^2 = 2, above the plane z = 0, and below the plane z = y + 3. Convert z = 2 Squareroot x^2 + y^2 to spherical coordinates. Find the volume of the region above the hemisphere rho = 1, z greaterthanorequalto 0, and below the cardioid of revolution rho = 1 + cos phi. Find the volume of the region inside the sphere x^2 + y^2 + z^2 = 2 and outside the cylinder x^2 + y^2 = 1.

Explanation / Answer

1) in cylindrical coordinates

x=rcos, y=rsin, z =z ,x2+y2=r2

y =x2

rsin=(rcos)2

rsin=r2cos2

r=sin/(cos2)

r=(1/cos)(sin/cos)

r=sectan

2)

in cylindrical coordinates

x=rcos, y=rsin, z =z ,x2+y2=r2

r =z/2

(x2+y2)= z/2

(x2+y2)= z2/4

4x2+4y2= z2

4x2+4y2-z2=0

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