Calculus: Analytic Geometry and Calculus, with Vectors

4.4 Areas and integrals 243

and this is equivalent to the relation

!6)

At

Jim 09 = 1 - t2 = 1/1 - sine B = cos 6

ne-.o

or to the firstof the formulas

(7) dBsin0=coscosdB^0 -sing.

The second follows from the first and the calculation

(8) dxcos 0 = dx sin (2-O)=- cos (2 -6)= - sin B.

21 Show how formulas of the preceding problem can be used to obtain the
integration formulas

f a=- x2dx=-Ix a2-x2+ a2 sin x+c

a

f

1/az1- x2

dx = sins a + c.

Then keep in contact with the external world by finding these formulas in your
book of tables.
22 Sketch a few figures which illustrate applications of the following fact.
If f is integrable (and hence bounded) over a 5 x < b, we can choose a positive
constant B such that f(x) + B > 0 when a 5 x <- b and write the formula

lbf(x) dx=fab [f(x) + B] dx-fab B dx,

which shows that fabf(x) dx is the result of subtracting the area of a rectangle

from the area of the set of points (x,y) for which a 5 x <--_ b and -B 5 y < f(x).

Remark: This problem and the next provide ways of reducing questions involving

integrals to questions involving integrals with nonnegative integrands.

23 Sketch a few figures which illustrate applications of the following fact.

If f is integrable (and hence bounded) over a 5 x S b, so also are the functions

g and Is defined by

g(x) _

1f(x)I

2

f(x) h(x) =If(x)I 2 f(x)

Moreover, g(x)? 0, h(x) Z 0, f(x) = g(x) - h(x), If(x)I = g(x) + h(x), and

f.

b

AX) dx =f ab g(x) dx- f.'h(x) dx

fab If (x) I dx=f ab g(x) dx + f ab h(x) dx.