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I am wondering if someone can help clarify the following? Suppose that I’m asked to find the volume of a cone. So, the volume would be ∫∫∫dxdydz or using polar coordinates ∫∫∫rdzdrdθ. Therefore the volume would be if I have a cone with base radius R and height H, I can express the radius r at any height z using similar triangles. I would get that r = Rz/H, where z is the height at any point in the cone.
Hence the volume would be ∫∫∫rdzdrdθ where Rz/H ≤ r ≤ R, 0 ≤ z ≤ H, and 0 ≤ θ ≤ 2π.
But I’m wondering why I can’t express the volume as ∫∫∫rdzdrdθ where 0 ≤ r ≤ Rz/H, and 0 ≤ z ≤ H and 0 ≤ θ ≤ 2π? Thanks!!
Hence the volume would be ∫∫∫rdzdrdθ where Rz/H ≤ r ≤ R, 0 ≤ z ≤ H, and 0 ≤ θ ≤ 2π.
But I’m wondering why I can’t express the volume as ∫∫∫rdzdrdθ where 0 ≤ r ≤ Rz/H, and 0 ≤ z ≤ H and 0 ≤ θ ≤ 2π? Thanks!!