Post on 25-Jan-2021
transcript
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%