Second portion of Multivariable Problem Set
Please show all work and draw figure
1) Every point on a cone is some distance from the axis of the cone. Find the average distance between all points on the surface of a circular cone (not including the base) and the axis of the cone. Assume that the cone has max radius a and height h.
a) Set up the equation of a cone that you will use to analyze the problem
b) What is function that gives the distance of a point on the surface of the cone, to the axis of the cone?
c) Identify the domain of the above function as it pertains to this problem. i.e. represent the domain using an equation?
c) Set up the integral to find the average value of the above function.
d) Evaluate the above integral. Draw a figure.
2) Find the mass M of a spherical planet of outer radius R if the density d of the planet at a radial distance ρ from the center is d = ρ + 1. Note that the density at ρ = 0 is ∞, still M is finite.
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