Calculus: Analytic Geometry and Calculus, with Vectors

13.1 Iterated integrals 655

3 By evaluating all of the integrals involved, show that

Jot dz fozf(Y) dy= Jo` (t - y)f(y) dy

when

(a) p > -1 and f(Y) = Yp

(b) k > 0 and f (y) = e -4y

(e) w 0 and f(y) = sin wy

4 The formula

Jot

udv=uv]o-Jotvdu,

which abbreviates the formula

fo u(x)v'(x) dz = u(z)v(x)]x=0-f of v(x)u'(x) d-,

for integration by parts, has unexpected applications. Assuming that f is con-

tinuous and

I f,,'d, foxf(Y)dY,

find the result of integrating by parts with

u(x) = foxf(y) dy v'(x) = 1

U, (x) = AX), v (x) _ - (t - x).

5 Calculate the two integrals I and j defined by

I - Jo° dx

foxf(x,Y) dy,

j

=foa dy Jva f(x,Y)

and show that they are equal when

(a) p > -1, q > -1 and f(x,Y) = xpy4

(b) f(x,y) = ex+v

6 Show that, when n > -1,

i i 2,.+s _ 2

Jo dxf (x+Y)"dy= (n+1)(n+2)

7 Show that

8 Show that

f dxJlx-f ydy=Zlog2.

1

J

i i 1 _

o dxJo (x+Y)Zdy 0D

dx

9 Supposing that 0 < p < 2 and p 0 1, showthat

i i 1 2z" - 2

Jo dxJo (x+y)pdy (2-p)(1-p)