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# Three Problems Involving Multiple Integrals

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Q1a)Find The area of the paraboloid x^2+y^2=z inside the cylinder x^2+y^2=9

b)write a triple integral in cylindrical coordinates for the volume inside the cone

z^2=x^2+y^2 and between the planes z=1 ans z=2

c)Find the moment of inertia of a circular disk (uniform density) about an axis

through its center and perpendicular to the plane of the disk

##### Solution Summary

We solve three problems in multivariable calculus involving multiple integrals.

##### Solution Preview

a) The area is given by the double integral

A = integral_S(dA) = integral_0^{2 pi}(integral_0^3(rho sec(psi(rho)) drho dphi))

where rho = sqrt(x^2 + y^2) is the cylindrical radial coordinate, phi is the azimuthal angle, and psi(rho) is the inclination angle of the paraboloid at a given point (which depends only on rho). Now the equation for the paraboloid in cylindrical coordinates is

z = rho^2

so ...

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