Use polar coordinates to find the volume of the given solid.?

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First, sketch it. Note that this is a cone. Assuming you want the volume inside it:

The element of volume is dx dy dz = r dr dθ dz in polar coords. So, evaluate


∫∫∫ r dr dθ dz.

Now for the ranges. θ obvious. r & z depend on each other.
Note that r = √(x² + y²) thus your cone is z = r. That is, given r, z goes from r to √81 = 9.

2π..9......9
∫......∫......∫ r dz dr dθ. Can you take it from there?
0....0......r...Show more