# Exercise

Find the length of the curve.

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2. ans

3. ans

4. ans

5. ans

6. ans

7. ans

8. ans

9. ans

10. ans

1. Find the surface area generated when the curve is revolved about the x-axis.

1. \lbrack ans:]

2. [ans:8]

3. [ans:]

4. [ans:]

# CENTROID OF A REGION; PAPPUS'S THEOREM ON VOLUMES

## The Centroid of a Region

Suppose that we have a thin distribution of matter, a plate, laid out in the -plane in the shape of some region . If the mass density of the plate varies from point to point, then the determination of the center of mass of the plate requires double integration. If, however, the mass density of the plate is constant throughout , then the center of mass of the plate depends only on the shape of and falls on a point that we call the centroid of . Unless has a very complicated shape, we can calculate the centroid of by ordinary one-variable integration.

We will use two guiding principles to find the centroid of a region . The first is obvious. The second we take from physics.

Principle 1: Symmetry. If the region has an axis of symmetry, then the centroid lies somewhere along that axis. (It follows from Principle 1 that, if the region has a center, then that center is the centroid.)

Principle 2: Additivity. If the region, having area consists of a finite number of pieces with areas and centroids , then

and

We are now ready to bring the techniques of calculus into play. Let's denote the area of by . The centroid of can be obtained from the following formulas:

To derive these formulas we choose a partition

of This breaks up into subintervals. Choosing as the midpoint of , we form the midpoint rectangles .The area of is and the centroid of is its center By Principle 1, the centroid of the union of all these rectangles satisfies the following equations:

(Here represents the area of the union of the rectangles.) As , the union of rectangles tends to the shape of and the equations we just derived tend to the formulas

## Problem 1

Find the centroid of the quarter-disc

Solution.

The quarter-disc is symmetric about the line We know therefore that Here

Since

The centroid of the quarter-disc is the point

## Problem 2

Find the centroid of the right triangle with vertices

Solution

There is no symmetry that we can use here. The hypotenuse lies on the line

Hence

and

Since we have

and

Let the region be between the graphs of two continuous functions and In this case, if has area and centroid then

Proof.

Let be the area below the graph of and let be the area below the graph of Then in obvious notation

and

Therefore

and

## Problem 3

Find the centroid of the region bounded by

Solution. Here there is no symmetry we can appeal to. We must carry out the calculations.

Therefore

# Pappus's Theorem on Volumes

All the formulas that we have derived for volumes of solids of revolution are simple corollaries to an observation made by a brilliant, ancient Greek, Pappus of Alexandria (circa 300 A.D.).

## PAPPUS'S THEOREM ON VOLUMES

A plane region is revolved about an axis that lies in its plane. If the region does not cross the axis, then the volume of the resulting solid of revolution is the area of the region multiplied by the circumference of the circle described by the centroid of the region:

where is the area of the region and is the distance from the axis to the centroid of the region.

As special cases of Pappus's Theorem we have

1. The Washer Method Formula. If and if the region bounded by and is revolved about the -axis, the resulting solid has volume

2. The Shell Method Formula. If and if the region bounded by and is revolved about the -axis, the resulting solid has volume

Note that

and

## Problem 4

Find the volume of the solid generated by revolving this region bounded by

Solution

Here and . Hence

## Problem 5

Find the volume of the doughnut (called torus in mathematics) generated by revolving the circular disc

about (a) the -axis, (b) the -axis.

Solution

The centroid of the disc is the center This lies units from the -axis and units from the -axis. The area of the disc is Therefore

(a)

(b)

## Problem 6

Find the centroid of the half-disc

by appealing to Pappus's theorem.

Solution. Since the half-disc is symmetric about the -axis, we know that All we need to find is .

If we revolve the half-disc about the x-axis, we obtain a solid ball of volume . The area of the half-disc is By Pappus's theorem

Simple division gives .

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