151: How to decide what change of variablesto use?Unfortunately , no hard and fast rule for this , butsometimes there is a "natural choice

"

.

e.g . b/c it makes the regionmuch

simplerand/or it makes the integrand much

simpler.

Example'tStewart 19.9.25 :

Arcos Hy¥) DAttrapezoid bdd by

11,0 ) , 12,01 , lo, 2), 10, l ) .

Solution : Natural choice for u and visu -- y

- x x-- ytx

Solve : x -- tzu y -- tty

Etihad'jwHhI:s¥ii.e. region of integral' 'm' cosftp.#dxdylNextpagdc.sfI).PYiu7!ldudv-

Hi:÷.

':* '.|,

cos It dude

-

The most systematic way of switching bounds isto describe the region completely algebraically

lie . using inequalities) and then substitute . .'

÷.

③① XZO ZO

② xtyl nuns 421③ g20 tty> O

④ xtys2 VEZ-

/! 'zoos Hi dude,

T gives x and yin terms of u and x

Ajamdude, a ftp.wdxdy

/! ! dxdyX-- UH

y-- u -y

Exercise : rewrite the integral in da dv

integration order(may need multipledouble integrals )-

n-

I Jacobian dell -- 2so the integrand is2- dudeI - lutvllu-ill

What happens to bounds :-

oexsl /,

om '→

"" iii. in .= . . . . . =T%