5.2

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5.2 _Volumes

Volumes

In trying to find the volume of a solid we face the same type of problem as in finding areas.

We have an intuitive idea of what volume means, but we must make this idea precise by using calculus to give an exact definition of volume.

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Volumes

We start with a simple type of solid called a cylinder (or, more precisely, a right cylinder).

As illustrated in Figure 1(a), a cylinder is bounded by a plane region B₁, called the base, and a congruent

region B₂ in a parallel plane.

The cylinder consists of all points on line segments that are perpendicular to the base and join B₁ to B₂. If the area of the base is A and the height of the cylinder (the distance from B₁ to B₂) is h, then the volume V of the cylinder is defined as V = Ah.

Figure 1(a)

Volumes

In particular, if the base is a circle with radius r, then the cylinder is a circular cylinder with volume V = πr²h [see

Figure 1(b)], and if the base is a rectangle with length l and width w, then the cylinder is a rectangular box (also called a rectangular parallelepiped) with volume V = lwh

[see Figure 1(c)].

Figure 1(b) Figure 1(c)

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Volumes

For a solid S that isn’t a cylinder we first “cut” S into pieces and approximate each piece by a cylinder. We estimate the volume of S by adding the volumes of the cylinders. We

arrive at the exact volume of S through a limiting process in which the number of pieces becomes large.

We start by intersecting S with a plane and obtaining a plane region that is called a cross-section of S.

Volumes

Let A(x) be the area of the cross-section of S in a plane P_x perpendicular to the x-axis and passing through the point x, where a ≤ x ≤ b. (See Figure 2. Think of slicing S with a

knife through x and computing the area of this slice.)

The cross-sectional area A(x) will vary as x increases from a to b.

Figure 2

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Volumes

Let’s divide S into n “slabs” of equal width ∆x by using the planes P_x

1, P_x

2, . . . to slice the solid. (Think of slicing a loaf of bread.)

If we choose sample points x_i∗ in [x_{i– 1}, x_i], we can

approximate the i th slab S_i (the part of S that lies between the planes P_x

i – 1 and P_x

i) by a cylinder with base area A(x_i∗) and “height” ∆x. (See Figure 3.)

Figure 3

Volumes

The volume of this cylinder is A(x_i∗) ∆x, so an

approximation to our intuitive conception of the volume of the i th slab S_i is

V(S_i) ≈ A(x_i∗) ∆x

Adding the volumes of these slabs, we get an

approximation to the total volume (that is, what we think of intuitively as the volume):

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Volumes

This approximation appears to become better and better as n → . (Think of the slices as becoming thinner and

thinner.)

Therefore we define the volume as the limit of these sums as n → .

But we recognize the limit of Riemann sums as a definite integral and so we have the following definition.

Volumes

When we use the volume formula it is

important to remember that A(x) is the area of a moving

cross-section obtained by slicing through x perpendicular to the x-axis.

Notice that, for a cylinder, the cross-sectional area is constant: A(x) = A for all x. So our definition of volume

gives ; this agrees with the formula

V = Ah.

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Example 1

Show that the volume of a sphere of radius r is .

Solution:

If we place the sphere so that

its center is at the origin, then the plane P_x intersects the sphere in a circle whose radius (from the Pythagorean Theorem) is

(See Figure 4.) So the cross-sectional area is

A(x) = πy²

Figure 4

= π(r² – x²)

Example 1 – Solution

Using the definition of volume with a = –r and b = r, we have

(The integrand is even.)

cont’d

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Volumes

Figure 5 illustrates the definition of volume when the solid is a sphere with radius r = 1.

From the result of Example 1, we know that the volume of the sphere is π, which is approximately 4.18879.

Figure 5

Approximating the volume of a sphere with radius 1

Volumes

Here the slabs are circular cylinders, or disks, and the three parts of Figure 5 show the geometric interpretations of the Riemann sums

when n = 5, 10, and 20 if we choose the sample points x_i∗ to be the midpoints

Notice that as we increase the number of approximating

cylinders, the corresponding Riemann sums become closer to the true volume.

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Volumes

The solids are all called solids of revolution because they are obtained by revolving a region about a line. In general, we calculate the volume of a solid of revolution by using the basic defining formula

and we find the cross-sectional area A(x) or A(y) in one of the following ways:

•

A = π(radius)²

Volumes

• If the cross-section is a washer, we find the inner radius r_in and outer radius r_out from a sketch (as in Figure 10) and compute the area of the washer by subtracting the area of the inner disk from the area of the outer disk:

A = π(outer radius)² – π(inner radius)²

Figure 10