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# Find the mass of the given shape

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1) (From prob_4.doc) Find the mass of the annulus (donut shape) having radius 1<r<2 when the density funciton rho(r,theta) = (1-ar^2) where a is a constant.

2) (From prob_5.doc) Find the total mass of the 3D object when the mass density rho and the object size is rho(r,phi,theta)=r^2sin(theta) where 1<r<2, 0<phi<pi and 0<theta<pi.

See the attached files.

https://brainmass.com/math/geometric-shapes/find-mass-shape-176737

#### Solution Summary

The solution finds the mass of the given shape.

\$2.19

## polar coordinates calculus

Use double integration in polar coordinates to find the volume of the solid that lies below the given surface and above the plane region R bounded by the given curve.

1. z=x^2+y^2; r=3

Evaluate the given integral by first converting to polar coordinates.

2. ∬_(0,x)^1,1▒〖x^2 dy dx〗

Solve by double integration in polar coordinates.

3. Find the volume of the solid bounded by the paraboloids z=12-2x^2-y^2 and
z=x^2+2y^2

Find the centroid of the plane region bounded by the given curves. Assume that the density is δ≡1 for each region.

4. x=-2, x=2, y=0, y=x^2+1

Find the mass and centroid of the plane lamina with the indicated shape and
density.

5. The region bounded by x=e, y=0, and y=ln⁡x for 1≦x≦e,with
δ(x,y)≡1

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