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PAGE: 523SET: ExercisesPROBLEM: 84
Please look in your text book for this problem Statement

The equations of the graphs are , ,  and .

(a)

Find the volume of the solid generated by revolving the region about the -axis.

The volume of the solid generated revolving about the - axis.

Formula for the volume of the solid with the Washer method,

.

The outer radius of revolution is .

The inner radius of revolution is .

.

Consider .

Solve the integral using integration by parts.

Formula for integration by parts:.

Here  and .

Consider .

Apply derivative on each side with respect to .

.

Consider .

Apply integral on each side.

.

Substitute corresponding values in .

.

Substitute  in .

The volume of the solid generated by revolving the region about the -axis is .

(b)

Find the volume of the solid generated by revolving the region about the -axis.

The volume of the solid generated revolving about the - axis is .

Here  and .

.

Solve the integral using integration by parts.

Formula for integration by parts:.

Here  and .

Consider .

Apply derivative on each side with respect to .

.

Consider .

Apply integral on each side.

.

Substitute corresponding values in .

The volume of the solid generated by revolving the region about the -axis is .

(c)

Find the centroid of the region.

Moments and center of mass of a planar lamina:

Let  and  be continuous functions such that  on , and consider the planar lamina of uniform density  bounded by the graphs of  and .

The moments about the -and -axes are ..

The center of mass  is  and , where  is the mass of the lamina.

Find .

Substitute ,  and  in .

.

Find .

Substitute ,  and  in .

Substitute .

.

Find .

Substitute ,  and  in .

.

Substitute .

.

Find the centroid.

Substitute  in .

.

Substitute  in .

.

The centroid of the region is .

(a) The volume of the solid generated by revolving the region about the -axis is .

(b) The volume of the solid generated by revolving the region about the -axis is .

(c) The centroid of the region is .



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