# Calculus

posted by Roy

Consider a hemispherical bowl with a radius of 12 cm. The bowl contains water to a depth of 9 cm. The density of the water at any point x cm below the surface of the water is given by 2e^(0.15x) g/cm^3.

A. Find the volume of the water in the bowl.
B. Find the mass of the water in the bowl.
C. Suppose that the bowl is covered and then inverted and that no water is lost in the process. Find the depth of the water in the inverted bowl.

1. Steve

A. The volume of a spherical cap of height h of a sphere of radius r is
V = π/3 h^2(r-h)
This is a known formula, but you can always verify it by finding the volume made by rotating a circular segment. For example,
V = ∫[h,r] π(r^2-y^2) dy

B. The mass of any small disc of water is its area times its density.
So, integrating again, with x=r-y,
M=∫[h,r] π(r^2-y^2) *2e^(0.15(r-y)) dy

C. Subtract the volume for (A) from 2π/3 r^3 and solve for h in the volume formula.

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