4.4.18 · D1Changing order of integration
Ek double integral har ek tiny tile par f Order of integration change karna bas usi exact set of tiles ko unke left/right edges se re-describe karna hai — top/bottom edges ki jagah (ya ulta) — isliye kuch bhi flip karne se pehle, humein un symbols mein fluent hona chahiye jo ek region aur ek summed-up total ko describe karte hain.
Yeh page har ek symbol banata hai jis par parent note "Changing order of integration" rely karta hai. Agar neeche koi symbol naya lage, toh uske teen parts padho: plain words → picture → kyun zaroorat hai. Koi cheez use nahi hogi jab tak ban na jaye.
Definition Ek point ek address hai
x sideways door: positive x x x = 0 y upar ya neeche : positive y y y = 0 ( x , y ) x y
Picture: do rulers right angles par jode hue, har ek ruler crossing point ke ek taraf se negative numbers mein aur doosri taraf positive numbers mein jaata hai. Flat sheet ka har dot — left ya right, upar ya neeche — ka exactly ek address ( x , y )
Topic ko yeh kyun chahiye: poora subject ek flat region ke baare mein hai, aur ek region bas "kaun se addresses ( x , y ) x y
Definition Ek curve donon numbers ke beech ek promise hai
y = g ( x ) mujhe x y " Letter g g ( x ) = x 2 x = h ( y ) mujhe height y x " Wahi curve, lekin doosri taraf se solve kiya hua .
Picture: plane par ek wiggly line drawn. Use y = g ( x ) left-to-right scan karna; usi line ko x = h ( y ) bottom-to-top scan karna.
Intuition DONO readings kyun matter karti hainOrder change karne ka poora trick yeh hai ki ek curve ke do naam hote hain. Origin se jaane wali line y = x bhi hai aur x = y bhi . Parabola y = x 2 bhi hai aur x = y bhi . Jab hum integration order flip karte hain toh humein boundary curve ko naye inner variable ke liye re-solve karna hota hai — yahi literally g h
Common mistake Square root ki kaun si branch?y = x 2 x x = + y ya x = − y Fix: ek point test karo. Agar region positive x x = + y
Common mistake Har boundary ek singley = g ( x ) x = h ( y ) full circle x 2 + y 2 = 1 x x y sideways-opening parabola x = y 2 y x x = h ( y ) y = g ( x ) fail karta hai. Fix: aisi boundary ko un pieces mein split karo jo ek genuine single-valued curve hain (upper arc y = + 1 − x 2 y = − 1 − x 2 region ko sub-regions mein split karo jo individually Type I ya Type II hain. Aise regions jo dono axes ko straddle karte hain unhe aksar yeh piecewise treatment chahiye — dekho Double Integrals over General Regions aur (round boundaries ke liye) Polar Coordinates in Double Integrals .
≤ a ≤ x ≤ b x a b a ≤ x ≤ b , g 1 ( x ) ≤ y ≤ g 2 ( x ) , x aur jinka y g 1 g 2
Picture: char inequalities char fences hain. Region R
Topic ko yeh kyun chahiye: ek integral ki limits yahi inequalities hain disguise mein. Parent note ki 5-step recipe hai "limits ko inequalities mein badlo, sketch karo, inequalities ko doosri taraf se re-read karo."
Common mistake Outer limit kabhi variable nahi hoti∫ 0 x ( ⋯ ) d x x x Sabse bahar wali slot mein numbers hone chahiye.
Definition Usi yard ko slice karne ke do tarike
Type I (vertically simple): vertical strips mein slice karo. x y d y d x Type II (horizontally simple): horizontal strips mein slice karo. y x d x d y
Picture: wahi triangle, ek baar vertical matchsticks mein shredded, ek baar horizontal matchsticks mein. Wahi triangle, alag matchsticks.
Topic ko yeh kyun chahiye: "order change karna" = "vertical matchsticks se horizontal ones mein switch karna." Poore patio par dust specks ki count dono taraf se same hoti hai — yahi deep reason hai ki answer kabhi nahi badalta.
Recall Differential order ko zor se padhoInner differential woh variable hai jo ek single strip ke andar move karta hai; outer differential woh variable hai jo tumhe strip-to-strip le jaata hai.
d y d x y x d x d y x y
Definition Ek sum banana, phir tiles ko zero tak squeeze karna
R Δ x Δ y Har tile par ek point pick karo aur f ( x , y ) number jo tile "carry" karta hai (dust density, mass, height...).
Tile ki area Δ x Δ y har tile add karo : i , j ∑ f ( x i , y j ) Δ x Δ y
Tiles ko kuch nahi kar do. Sum ek single number par settle ho jaata hai, double integral :
∬ R f d A = lim ∑ i , j f Δ x Δ y .
Symbols:
∫ Sum ke liye "S", tall stretched.∬ d A A rea; coordinates mein d A = d x d y = d y d x f ( x , y )
Topic ko yeh kyun chahiye: kyunki ek sum ko rows-then-columns ya columns-then-rows mein total kiya ja sakta hai aur same total milta hai, isliye dono iterated integrals ek hi ∬ R f d A
Worked example Unit triangle par ek concrete totalMaano R 0 ≤ x ≤ 1 , 0 ≤ y ≤ x f = 1 f = 1 ∫ 0 1 ∫ 0 x 1 d y d x = ∫ 0 1 x d x = 2 1 . x x = y x = 1 ∫ 0 1 ∫ y 1 1 d x d y = ∫ 0 1 ( 1 − y ) d y = 1 − 2 1 = 2 1 . 2 1
Definition Derivative ko undo karna∫ e y 2 d y e y 2 e y 2 y s i n y usual functions se express hi nahi ho sakta — dekho Improper and Non-elementary Integrals .
Topic ko yeh kyun chahiye: agar inner integral ek order mein closed form mein nahi hai, toh flip karne par ek inner integral mil sakta hai jo hai. Yahi parent note mein motivation #1 hai.
Worked example Magic cancellation, poore context mein dikhaya gayaMaano poora iterated integral unit triangle par d x d y ∫ y = 0 1 ( ∫ x = 0 y y s i n y d x ) d y . Inner integral par focus karo. Yahan x dummy variable hai — yeh woh letter hai jo strip mein move karta hai aur integration ke baad gayab ho jaata hai. Quantity y s i n y koi x hai, isliye inner ∫ d x constant hai aur simply bahar aa jaata hai. Upper limit y strip length hai — yeh outer variable y x ∫ x = 0 y y s i n y d x = y s i n y [ x ] x = 0 x = y = y s i n y ⋅ y = sin y . x y y 1 ∫ 0 1 sin y d y Yahi cancellation reordering ka poora payoff hai.
Neeche diagram top to bottom padho: har box is page ka ek foundation hai, aur har arrow ka matlab hai "upar wala idea pehle chahiye neeche wala samajhne se pehle." Koi bhi path neeche bottom box "Changing order of integration" tak trace karo — woh path ek valid study order hai. Agar koi box shaky lage, aage badhne se pehle uske section par wapas jaao. (Agar diagram tumhare reader mein render nahi hota, wahi dependencies Equipment checklist questions mein spell out hain jo follow karte hain.)
Curves y equals g and x equals h
Inequalities carve a region R
Type I vertical strips and Type II horizontal strips
Sum of tiles becomes double integral
Fubini both orders equal same total for nice f
Changing order of integration
Re-solve boundary for new inner variable
Antiderivatives elementary or not
Khud test karo — sirf tab reveal karo jab tum zor se jawab de chuke ho.
Address ( x , y ) x x < 0 x > 0 y y < 0 y > 0
Origin se 4 5 ∘ y = x x = y
Positive x y = x 2 x x = + y (sirf positive branch).
Puri circle x 2 + y 2 = 1 y = g ( x ) Har
x do
y -values deta hai (upper aur lower arcs); tumhe use
y = + 1 − x 2 aur
y = − 1 − x 2 mein split karna hoga ya region split karna hoga.
∫ a b ∫ g 1 g 2 f d y d x Outer wali a , b x
Ek vertical strip kaun se differential order se correspond karta hai? d y d x y x
∬ R f d A ∑ f Δ x Δ y R f
Ek aise hypothesis ka naam batao jiske under Fubini order swap karne deta hai. f R f ≥ 0 ∬ ∣ f ∣ < ∞
Order flip karne ki zaroorat hi kyun hai? Inner integral ek taraf elementary antiderivative nahi ho sakta lekin doosri taraf trivial ho sakta hai.
Jab tum ek strip 9 0 ∘ Use naye inner variable ke liye re-solve karo (g h
Aage: in symbols ke secure hone ke baad, parent note Changing order of integration inhe 5-step reorder recipe mein turn karta hai. Related deep tools Double Integrals over General Regions , Polar Coordinates in Double Integrals aur Change of Variables (Jacobian) mein milte hain.
