# polar coordinates

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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=lnx for 1≦x≦e,with

δ(x,y)≡1

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This solution utilizes double integration in polar coordinates.

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Problems:

1. 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.

2. Evaluate the given integral by first converting to polar coordinates.

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

4. Find the centroid of the plane region bounded by the given curves. Assume that

the density is =1 for each region.

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

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

density. The region bounded by x = e, y = 0, and y = ln x for 1xe, with (x,y)=1

Solutions:

1. The general formula for calculation of volumes bounded by given surfaces is:

(1)

The plane region R where the curve r = ...

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