Multivariable Calculus: find vol enclosed by z=0, z=2+x^2+y^2, r=cos(theta)

[Sam202]

I am asked to calculate the volume enclosed by the following equations:

z =0
z = 2 + x² + y²
r = Cos(theta)

I know that the second equation can be changed to its polar form: (if relevant to this problem)

z = 2+ r²

I also know that the formula to calculate said volume is:

V = Double Integral (z(x,y)) or alternatively..... V = Double Integral (r*dr d(theta))

However, I'm lost in terms of figuring out the limits of integration and the integral I need to calculate in order to obtain said volume. Any help would be appreciated.

 

[Deleted member 4993]

I am asked to calculate the volume enclosed by the following equations:

z =0
z = 2 + x² + y²
r = Cos(theta)

I know that the second equation can be changed to its polar form: (if relevant to this problem)

z = 2+ r²

I also know that the formula to calculate said volume is:

V = Double Integral (z(x,y)) or alternatively..... V = Double Integral (r*dr d(theta))

However, I'm lost in terms of figuring out the limits of integration and the integral I need to calculate in order to obtain said volume. Any help would be appreciated.

z = 2 + x² + y²

This is an inverted (truncated) cone.

Make an approximate sketch.

What shape would you get from r = Cos(Θ)?

Where does these shapes intersect?

Please tell us about your findings....

 

[Sam202]

z = 2 + x² + y²

This is an inverted (truncated) cone.

Make an approximate sketch.

What shape would you get from r = Cos(Θ)?

Where does these shapes intersect?

Please tell us about your findings....

In polar coordinates, I know r =cos(theta) is graphically a circle with radius = (1/2) and center in (1/2,0) with respect to the horizontal axis; however, I'm new to polar coordinates; and therfore am confused as to how i can relate both "r" and "z" equations given they both belong to a different coordinated system.

 

[Ishuda]

In polar coordinates, I know r =cos(theta) is graphically a circle with radius = (1/2) and center in (1/2,0) with respect to the horizontal axis; however, I'm new to polar coordinates; and therfore am confused as to how i can relate both "r" and "z" equations given they both belong to a different coordinated system.

You have a 3D system and, IMO, it would be better to convert everything to Cartesian coordinates:
(1) z = 0, a plane
(2) z = 2 + x² + y², an infinite paraboloid opening up
(3) (x-\(\displaystyle \frac{1}{2}\))² + y² = \(\displaystyle \frac{1}{4}\), an infinite cylinder

or, if you would like, translate to a cylindrical coordinate system:
(1) z = 0, a plane
(2) z = 2 + r², an infinite paraboloid opening up
(3) r = cos(\(\displaystyle \theta\))

The plane [(1)] and cylinder [(3)] obviously intersect at z=0 but where does the paraboloid [(2)] and cylinder [(3)] intersect?