Calculus: Analytic Geometry and Calculus, with Vectors

4.4 Areas and integrals 241

16 Let .4 be the area of the circular disk of
radius a shown in Figure 4.493. Explain the
ideas associated with the calculation

11 =lim12axAx = 2r faxdx = -a2.

Hint: Think of the ring between the two inner
circles as being a ribbon of width Ax and length
21rx, the length being (by definition of a) the cir-
cumference of a circle of radius x.

Figure 4.493

17 Sketch graphs of sin x and cos x over the interval 0 < x < ,r and then,

with the aid of this information, sketch graphs of sin2 x and cos' x. Use these

graphs to obtain a reason why it should be true that

(1)

Note also that

(2)

o sin x dx = for cos2 x dx.

for (sin2 x + cos2 x) dx = for I dx = r.

What can we now conclude about the integrals in (1)? Taking a totally different

tack, use the formulas

(3)

1 - cos 2x l +cos 2x

sin2 x = 2 , cos2 x -^2

to evaluate the integrals in (1). Make all of the results agree.

18 Prove the formula

10a

7ra2

a2 - x2 dx = 4

by observing that the integrand is nonnegative and constructing a region of

which the integral is the area. Hint: Let y = a2 - x2 and, after tinkering

with this equation, draw an appropriate figure.

19 Let a and b be constants for which 0 < b < a. Show that if y >= 0,

0<x5 a, and

x.2 y2

(1) a2+b2= 1,

then

y =b-1/ a 2 - x2.

a

Let S be the set of points inside the graph of (1); as we shall learn later, thegraph

is an ellipse. With the aid of Figure 4.494 show

that

f

ISI a 1 o

a2 - x2 dx.

Figure 4.494

With the aid of the preceding problem, show that 6

BSI = Trab. This is a result that many people 41!a

remember: the area of a circular disk is ,raa and

the area of an elliptic disk is 7rab.