Changing order of integration: Question bank — Hinglish deep dive

4.4.18 · D5 · Maths › Multivariable Calculus › Changing order of integration

Shuru karne se pehle, ek shared vocabulary — aur ek picture jo ise anchor kare. Poore time, hum $R$ likhte hain region of integration ke liye: woh flat 2D patch jo $x y$ -plane mein hai aur jiske upar double integral $\iint_{R} f d A$ liya jaata hai. Har "strip" is $R$ ke andar hoti hai.

Outer = "main kitni dur sweep karoon"; inner = "yeh ek strip kahaan shuru aur khatam hoti hai". Figure bilkul yahi dikhata hai: outer range axis par flat interval hai, inner limits woh do curves hain jo red strip touch karti hai.

True or false — justify

TF1. Double integral ka order swap karna hamesha uski numerical value change karta hai.

False — Fubini's Theorem ke through dono iterated integrals same area-weighted total ke barabar hain; sirf bookkeeping (limits) badlti hai, number kabhi nahi.

TF2. Agar inner integral ka koi elementary antiderivative nahi hai, toh double integral bilkul evaluate nahi ho sakta.

False — tum ek order mein atke ho sakते ho, lekin reordering inner integral ko kuch trivial mein badal sakta hai (e.g.

e^{y^{2}}

inner swap ke baad

y e^{y^{2}}

ban jaata hai), isliye double integral ka phir bhi ek clean value hota hai.

TF3. Swap karne ke baad, naye outer limits sirf old limits ke charon numbers ke min aur max hain.

False — outer limits new outer variable ki overall range hain, jo tum

R

ki sketch se padhte ho, purane numbers ko idhar-udhar shuffle karke nahi.

TF4. Kisi bhi region ke liye, vertical-strip description aur horizontal-strip description dono ek single integral ke roop mein (bina splitting ke) exist karte hain.

False — bahut saare regions ek taraf se "simple" hote hain lekin doosri taraf se do ya zyada integrals mein split karne padte hain; sirf convex-ish, well-behaved regions dono taraon se single-integral hote hain.

TF5. Jab tum order flip karte ho, region ki boundary curves same rehti hain.

True curves ke roop mein, lekin tum har equation ko naye inner variable ke liye re-solve karna chahiye — e.g.

y = x^{2}

same parabola rehti hai lekin

x = y

likhi jaati hai jab

x

inner ban jaata hai.

TF6. Fubini's theorem ko guarantee karne ke liye ki dono orders agree karein,

f

ka positive hona zaroori hai.

False — region

R

par

f

ki continuity (ya integrability with a finite integral of

∣ f ∣

over

R

) woh hai jo require hota hai; swap valid hone ke liye

f

ka sign irrelevant hai.

TF7. Agar ek region par ek boundary

x = y

hai, toh swapping legitimately

x = - y

bhi introduce kar sakta hai.

Tabhi jab region actually negative branch include karta hai — tum woh branch choose karte ho jo

R

mein lie karta hai ek interior point test karke, dono automatically nahi.

TF8. Order

d x d y

hamesha horizontal strips se correspond karta hai.

True — innermost differential us variable ko name karta hai jo ek single strip ke andar move karta hai, aur

d x

move hona matlab strip horizontal hai (fixed

y

TF9. Reordering ek proper integral ko improper mein badal sakta hai.

True — kuch Improper and Non-elementary Integrals ke liye region unbounded hota hai ya

f

blow up karta hai, aur ek orientation singularity ko ek finite inner integral ke roop mein chupa sakta hai jabki doosri usse expose karti hai; swap wahan bhi valid hai jahan Fubini's hypotheses hold karti hain.

Spot the error

SE1. "Reverse

\int_{0}^{1} \int_{0}^{x} f d y d x

to get

\int_{0}^{x} \int_{0}^{1} f d x d y

." — Kya galat hai?

Outer limit

\int_{0}^{x}

mein abhi bhi

x

hai (Blunder A); outer limits constants hone chahiye. Us triangle par sahi swap hai

\int_{0}^{1} \int_{y}^{1} f d x d y

SE2. "Region hai

x^{2} \leq y \leq x

, toh swap karne ke baad inner limits hain

\int_{x^{2}}^{x} f d x

— yani abhi bhi

x

ke terms mein likha hai." — Kya galat hai?

Boundaries ko

x

ke liye kabhi re-solve nahi kiya gaya (Blunder B). Solve karo

y = x^{2} \Rightarrow x = y

aur

y = x \Rightarrow x = y

, jisse inner limits milti hain

\int_{y}^{y} f d x

aur outer

y \in [0, 1]

SE3. "

y = x

aur

y = x^{2}

ke beech wale lens mein fixed

y \in (0, 1)

par strip ke liye,

x

runs from

y

y

." — Kya galat hai?

Order ulta hai:

0 < y < 1

ke liye

y < y

hota hai, isliye strip

x = y

par enter karti hai aur

x = y

par exit karti hai; lower limit smaller value honi chahiye,

x = y

SE4. "Maine saari limits ka min (0) aur max (1) outer ke liye copy kar liya, toh outer

0

1

hai — koi sketch ki zaroorat nahi." — Kya galat hai?

Sketch skip karna (Blunder C) — asli outer range aur region ko split karne ki zaroorat hai ya nahi, yeh sirf geometrically dekha ja sakta hai; ek lucky number match ise validate nahi karta.

SE5. "

\int_{0}^{2} \int_{0}^{4 - x^{2}} f d y d x

mein region ek full circle hai, toh main swap karke

\int_{- 2}^{2} \int_{0}^{4 - y^{2}} f d x d y

likhta hoon." — Kya galat hai?

Original region sirf first quadrant mein quarter disk hai (

x, y \geq 0

), full ya half circle nahi. Naye limits:

0 \leq y \leq 2

0 \leq x \leq 4 - y^{2}

SE6. "Inner limits constants ban gaye aur outer

y

ka function ban gaya — yeh ek valid swapped integral hai." — Kya galat hai?

Yeh ulta hai. Inner limits outer variable ke functions ho sakte hain; outer constant hone chahiye. Yeh ek ill-formed integral describe karta hai.

SE7. "

\int_{0}^{1} \int_{x}^{1} e^{y^{2}} d y d x

swap karte waqt main boundary ko

y = x

rakhta hoon aur

\int_{0}^{1} \int_{0}^{y = x} \dots

likhta hoon." — Kya galat hai?

Tumhe inner variable

x

explicitly express karna hoga (Blunder B):

y = x

x

ke liye solve karo taaki

x = y

mile, isliye inner hai

\int_{0}^{y}

, giving

\int_{0}^{1} \int_{0}^{y} e^{y^{2}} d x d y

SE8. "

\int_{1}^{e} \int_{0}^{l n x} f d y d x

ke region ke liye swap

\int_{0}^{1} \int_{1}^{e^{y}} f d x d y

deta hai, bina yeh soche ki

y = ln x

ke kis taraf hain." — Yahan kya sahi/galat hai?

Yeh actually sahi hai:

y = ln x

x

ke liye solve karna

x = e^{y}

deta hai (right edge), left edge

x = 1

hai, aur

y

0

1

tak range karta hai. Trap yeh assume karna hai ki yeh galat hai — hamesha suspicion ki jagah ek interior point test karke verify karo.

Why questions

WQ1. Outer limits constants kyun hone chahiye jabki inner limits curves ho sakte hain?

Outer integral woh aakhri wala hota hai jo kiya jaata hai — iska result ek plain number hona chahiye, isliye koi free variable survive nahi kar sakta; inner integral pehle kiya jaata hai jab outer variable fixed hota hai, isliye woh variable legitimately inner bounds mein appear kar sakta hai.

WQ2. Reordering kabhi kabhi ek "impossible" inner integral ko trivial kyun bana deta hai?

Reversing doosre variable ko andar dalta hai; strip-length factor jo woh contribute karta hai (e.g.

y

se multiply karna) exactly woh derivative factor supply kar sakta hai jo chahiye, jisse substitution ya cancellation antiderivative ko collapse kar deta hai.

WQ3. Region ko sketch karna "80% of the work" kyun kaha jaata hai?

Picture hi akela reliable tarika hai asli outer range padhne ka, required splittings spot karne ka, aur correct root branch choose karne ka — akela algebra silently wrong limits produce karta hai.

WQ4. Swap Fubini's theorem pe kyun rely karta hai, obvious kyun nahi hai?

Double integral 2D region

R

par ek single limit hai; ki yeh limit kisi bhi iterated (nested 1D) integral ke barabar hai, yeh ek genuine theorem hai jo integrability require karta hai, notation ki triviality nahi. Dekho Fubini's Theorem.

WQ5. Ek region ko ek orientation mein splitting ki zaroorat kyun pad sakti hai lekin doosri mein nahi?

Agar ek horizontal strip alag curves par exit kare apni height ke hisaab se (boundary "change formula" karti hai beech mein), tum split karo; vertical strips ki taraf turn karne se ho sakta hai har strip ek single curve par exit kare, koi split ki zaroorat nahi.

WQ6. Hum kabhi kabhi

d x d y

swap karne ki bajaye Polar Coordinates in Double Integrals kyun use karte hain?

Circular/annular regions ke liye dono Cartesian orders ugly

r^{2} - x^{2}

limits dete hain; polar region ko

(r, θ)

mein ek simple rectangle bana deta hai, "reordering" ka ek alag (aur aksar behtar) tarika via Change of Variables (Jacobian).

WQ7. Shared boundary curve swap mein special role kyun play karti hai?

Yahi woh single equation hai jo dono descriptions touch karti hain; poora order change usi shared boundary ko opposite variable ke liye solve karne ka algebraic act hai — baaki sab sirf yeh relabel karta hai ki kaunsa slot inner hai.

WQ8. Value is par depend kyun nahi karti ki hum rows-then-columns ya columns-then-rows sum karein?

Double integral same tiles ke set par ek sum ka limit hai, aur finite summation commutative hoti hai; iterated integrals sirf identical tiles add karne ke do orders hain, isliye totals coincide karte hain.

Edge cases

EC1. Jab region ek single point par degenerate ho jaata hai (do boundary curves milti hain), toh swapped limits kya hain?

Inner limits collapse ho jaati hain (lower = upper), strip ki length wahan zero ho jaati hai, aur integral us value par

0

contribute karta hai — ek measure-zero edge jo kabhi result affect nahi karta.

EC2. Swap karne ke baad

y = x

aur

y = x^{2}

ke beech wale lens mein

y = 0

aur

y = 1

par kya hota hai?

Dono endpoints par curves milti hain (

x = y = y

y = 0

aur

y = 1

par), isliye inner interval

[y, y]

ek point tak shrink ho jaata hai — region corners

(0, 0)

aur

(1, 1)

tak pinch ho jaata hai.

EC3. Agar region ek rectangle hai

a \leq x \leq b, c \leq y \leq d

, toh swap karne par kya badalta hai?

Sirf differential order — dono inner aur outer limits already constants hain, isliye

\int_{a}^{b} \int_{c}^{d} = \int_{c}^{d} \int_{a}^{b}

; yeh Fubini ka sabse clean case hai.

EC4. Ek region par jahan

y

appear karta hai, inner limits order karne se pehle yeh check kyun karna zaroori hai ki

0 < y < 1

hai ya

y > 1

Kyunki

y

line

y

ko cross karta hai:

0 < y < 1

ke liye

y < y

hai, lekin

y > 1

ke liye

y < y

; smaller value hamesha lower limit hoti hai, isliye inequality flip ho sakti hai aur enter/exit edges mislabel ho sakti hain.

EC5. Kya ho agar ek horizontal strip region se exit kare, re-enter kare, aur phir exit kare (ek non-simple region)?

Single inner integral invalid hai; tum strip ke

R

ke andar har contiguous piece ke liye ek integral ke sum mein split karo — ek case jo sketch turant reveal karta hai.

EC6. Ek unbounded region ke liye (e.g.

x \geq 0, y \geq x

out to infinity), kya swap phir bhi legal hai?

Haan jahan improper double integral absolutely converge kare; tab Fubini/Tonelli swap karne dete hain, lekin limits improper rakhni hain aur convergence confirm karni hogi — dekho Improper and Non-elementary Integrals.

EC7. Kya reordering idea teen variables tak extend hota hai?

Haan — Triple Integrals — Order of Integration ke liye

d x, d y, d z

3! = 6

possible orders hain, sab Fubini ke through barabar hain, aur same "outer constants, inner surfaces/curves" discipline har level par apply hoti hai.

Recall Ek-line survival rule

Upar ka har trap usi reflex se defeat hota hai: $R$ sketch karo, naye outer ki full range constants ke roop mein choose karo, har boundary ko naye inner variable ke liye re-solve karo, aur ek interior point test karo. Woh single rule jo is page par sabse zyada errors fix karta hai ::: Outer limits constants hone chahiye; inner limits curves ho sakte hain — kabhi ulta nahi.