Calculus: Analytic Geometry and Calculus, with Vectors

4.5 Volumes and integrals 249

(x,y,z) for which y 0, we make a partition of the interval0 < y 5 b. When

0 < y < b, we can put (2) in the form

(3)

and hence in the form

(4)

When y has the constant value yt*., (4) has the form

,.2 .2

!q2 + B2 = 1,

x2 z2 (^1) (b2-y2)
x2 2
(b b2 --Y2)

+(b b2 -

2)2

= 1.

where

(6) =b b2-Yk2, B=b b2-Yk2

This shows that, as Figure 4.593 indicates, the plane having the equationy = yk
intersects our solid in an elliptic disk
which, according to Problem 19 of
Section 4.4, has area TrdB or

(7)

aac

b2

(b2 Yk2)

The volume of the slab of our solid
which lies between the planes having
the equations y = yk_, and y = yk is

Figure 4.593

then exactly or approximately the result of multiplying (7) by oyk. Thus

(8)

rac

V = 2 lim I

b2

(b2 - Yk 2) AY."

the factor 2 being required because we partitioned only the interval 0 S y 5 b.

The limit of Riemann sums being a Riemann integral, we obtain

(9) v 2bcc fob (b2- y2) dy

and hence (1). In case two of the three numbers a, b, c are equal, say a = c,

the graph of (2) is called a spheroid. When finding the volume of the solid

bounded by a spheroid, it is possible to simplify matters by using circular disks

instead of elliptic disks. Some scientists consider it to be more fun to work out

the above formulas than to remember that a spheroid for which a = c < b

is called a prolate spheroid (like the surface of a cucumber or a watermelon) and

that a spheroid for which a = c > b is an oblate spheroid (like the surface of a

pancake or an unscarred earth that bulges at its equator and is flattened at its

poles because of its rotation).

(^18) From time to time, we recognize the fact that some scientific terminologies