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103

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PAGE: 535SET: ExercisesPROBLEM: 103
Please look in your text book for this problem Statement

The equations of the graphs are , and .

(a)

Find the area of the region.

Find the integral limits by equating two curve equations.

.

Area of the region bounded by , , and .

Consider and .

The two curves on interval .

The area enclosed by the curves is .

.

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 area of the region 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.

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

.

The outer radius of revolution is .

The inner radius of revolution is .

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 .

Again apply integration by parts.

.

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

(c)

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 .

(d)

Find the centroid of the region.

Moments and center of mass of a planar lamina:

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

The moments about the -and -axis are ..

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

Substitute , , and in .

Substitute .

.

.

Substitute , , and in .

Substitute .

.

.

The centroid of the region is .

(a) The area of the region is .

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

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

(d) The centroid of the region is .



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