The discussion focuses on calculating the volume between the cone defined by the equation z = √(x² + y²) and the plane z = 14 + x, above the disk x² + y² ≤ 1. Participants emphasize the importance of consistently using either Cartesian or cylindrical coordinates to avoid confusion. The cone's equation should be expressed as z = f(r) and the plane as z = g(r, θ) in polar coordinates, where the height of the volume is determined by the difference between these two functions. The integration limits are established as 0 ≤ r ≤ 1 and 0 ≤ θ ≤ 2π.
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dynamicsolo said:I think you are going to need to come down firmly on the side of working with either Cartesian or cylindrical coordinates, because this mixture of systems is creating unnecessary confusion.
You have an "apex-down" cone (the apex at ( 0, 0 ) ) opening "upward" in the positive z-direction being intersected by an oblique plane. You will find it easier to deal with this if you express the cone as z = f(r) (which will be the "floor" of your volume) and z = g( r, theta ) [what is 14 + x in polar coordinates?] (which is the "ceiling" of your volume).
You can now get an integrand function for the "height" of the volume as a function of r and theta . You will be integrating over 0 ≤ r ≤ 1 and you have the correct interval for theta. Be sure to use the infinitesimal volume elements for cylindrical coordinates. This will not be too hard to set up (it's a bit more work to actually integrate...).