Calculus: Analytic Geometry and Calculus, with Vectors

656 Iterated and multiple integrals

10 Supposing that n > -2 and is 76 1, show that

1 x

f

- 1

dx fo (x I Y)" dY =

2n+1

o (n + , 1) (n + 2)

Investigate the case in which n = -1.

11 Show that

r1 2xy^3

r1 y

(a) Jo dx

fx

xdy 4 (b)Jo dxJ= xdy =

/1 dxr2zx log2 1 (lx i

(`) fo x Y

dy

=^2 (d) o dx fx ydy

12 Show that making one integration gives the formula

1 x (^1) r11 1
2fo dxfo 1 - xydy = 2f xlogl x2dx
which, so far as we know, has dubious validity because the last integrand is
meaningless when x = 0 and when x = 1. Show that
f
1
o 1 -xydy
has the value 0 when x = 0 and does not exist (or has the value + oo) when
x = 1. Then proceed to the next problem.
13 The integrals in
(1) r1
x

2Jo dxf

1

0 1 -xydy

cannot exist as iterated Riemann integrals because the integral in

(2) f(x) = fox 11 xydy

does not exist as a Riemann integral when x = 1. However, when 0 5 x < 1,

(3) f(x) =

fox

(I + xY + x2y2 + x3y3 + -- -) dy

Cy +

xy2

'Y' "'

= 2 + 3 + 4 ,+.. ..]y=ob=x

L x3 x6 x7

+2+3 +4+...

To each positive integer is there corresponds a positive number S such that

(4)

f(x)x+23+3s+...+xn>(1++...

+n/-1

when 1 - S < x < I (why?) and hence (why?)

(5) lim f(x) = m.

x-+1-