Visual walkthrough — Double integrals over rectangles — Fubini's theorem

4.4.16 · D2 · Maths › Multivariable Calculus › Double integrals over rectangles — Fubini's theorem

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Step 1 — Hum measure kya kar rahe hain?

KYA. Hamaare paas ek flat rectangular floor hai aur uske upar ek curved roof tair raha hai. Hum woh space measure karna chahte hain jo floor aur roof ke beech mein trapped hai.

KYO. Jo bhi aage aata hai woh sirf is ek space ke blob ko measure karne ka ek clever tarika hai. Agar aap blob ko clearly dekh sako, toh baad ke har step mein sirf bookkeeping hai.

PICTURE. Figure dekho. Zameen par jo black rectangle hai woh hamaara floor hai. Uske corners char plain numbers se fix hain:

$R=\underset{\text{kitna door }x\text{ jaata hai}}{\underbrace{[a,b]}}\times \underset{\text{kitna door }y\text{ jaata hai}}{\underbrace{[c,d]}}$

• $x$ left-to-right chalta hai, number $a$ se number $b$ tak.
• $y$ front-to-back chalta hai, number $c$ se number $d$ tak.
• Upar tairta hua red surface hamaara roof hai, likha jaata hai $z=f(x,y)$: har floor point $(x,y)$ ke liye, wahaan ka roof height number $f(x,y)$ hai.

Hum joh poori quantity chahte hain woh red-capped solid ka volume $V$ hai.

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Step 2 — Volume ko tiny towers se approximate karo

KYA. Cleanly slice karne se pehle, hum volume ko chhote chhote rectangular columns se banate hain — yeh honest, brute-force definition hai.

KYO. Volume yahaan se aata hai. Hum floor ko ek grid of small tiles mein chop karte hain, har tile par ek tower khada karte hain jitna uske upar roof hai, aur phir towers ko add karte hain. Yeh wahi trick hai jo Riemann sums mein hai, bas intervals ki jagah tiles hain.

PICTURE. Har floor tile ki width $\Delta x$ aur depth $\Delta y$ hai, toh uska area hai

$\Delta A=\underset{\text{tile width}}{\underbrace{\Delta x}}\cdot \underset{\text{tile depth}}{\underbrace{\Delta y}}.$

Tile number $(i,j)$ ke andar koi bhi point $(x_i^{\ast },y_j^{\ast })$ chuno — star ka matlab sirf "us tile mein koi sample point" hai. Uske upar khade tower ka

$\text{tower volume}=\underset{\text{sample par height}}{\underbrace{f(x_i^{\ast },y_j^{\ast })}}\cdot \underset{\text{base area}}{\underbrace{\Delta A}}.$

Har tower add karo:

$S={\sum }_i{\sum }_{j}f(x_i^{\ast },y_j^{\ast })\Delta A.$

Figure mein red tower ek aisa hi term hai; grey towers uske neighbours hain.

Problem yeh hai: yeh ek double limit hai — seedha evaluate karna bahut brutal hai. Steps 3–6 ise do aasaan one-dimensional integrals mein badal dete hain.

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Step 3 — Ek direction freeze karo: ek single slice banao

KYA. Tiny towers ki jagah, ek bada chaaku lo aur solid ko ek flat vertical wall se kaato. $x$ ko kisi value $x_0$ par freeze karo aur solid ke andar straight back slice karo.

KYO. Us frozen wall par, $x$ ab ek variable nahi hai — yeh ek locked number hai. Toh roof, sirf is wall par dekha jaaye toh, ek plain 1-variable curve $y\mapsto f(x_0,y)$ hai. Aisi curve ek ordinary flat area bound karti hai, aur flat areas hum single-variable calculus se already jaante hain.

PICTURE. Figure mein red shaded face cut hai. Back-position $y$ par uski height $f(x_0,y)$ hai; yeh $y=c$ se $y=d$ tak failaa hua hai. Uska area — ise $A(x_0)$ kahein — us height ka ordinary integral hai:

$A(x_0)={\int }_c^d\underset{\begin{array}{c}\text{frozen wall ke}\\ \text{saath roof height}\end{array}}{\underbrace{f(x_0,y)}}dy.$

$dy$ ko padhein "sweep $y$ from $c$ to $d$, red face ki thin vertical strips add karte hue."

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Step 4 — Slice ko slide karo: area position ka function ban jaata hai

KYA. Ab $x$ ko unfreeze karo aur cutting wall ko left edge $x=a$ se right edge $x=b$ tak slide karne do. Har stopping point par face ka ek (shayad alag) area hoga.

KYO. Solid sirf ek slice nahi hai — yeh unka continuous stack hai. Poore stack ko describe karne ke liye humein jaanna hai ki jab wall slide karti hai toh slice-area kaise badalta hai. Isse hume ek variable ka ek naya single function milta hai.

PICTURE. Figure teen positions par sliding wall dikhata hai. Neeche, black curve hai

$A(x)={\int }_c^{d}f(x,y)dy,$

slice-area as a function of jahaan wall khadi hai. Red dot Step 3 ka $A(x_0)$ mark karta hai — is curve par sirf ek point. Jahan roof tall aur wide hai, $A(x)$ large hai; jahan solid pinch hota hai, $A(x)$ dip karta hai.

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Step 5 — Slices ko wapas volume mein stack karo

KYA. Har slice ko ek tiny thickness $dx$ do aur unhe side by side glue karke solid rebuild karo.

KYO. Ek slice jiska face-area $A(x)$ hai aur thickness $dx$ hai, woh volume $A(x)dx$ ka ek thin slab hai. $x=a$ se $x=b$ tak saare slabs add karna exact solid reassemble karta hai — yeh precisely Volume by slicing (single-variable) hai, "loaf of bread" wala idea, ek dimension upar.

PICTURE. Figure slabs se rebuild hua loaf dikhata hai; red slab position $x$ par wala hai, jis par thickness $dx$ visibility ke liye thick draw ki gayi hai.

$V={\int }_a^b\underset{\text{slab face area}}{\underbrace{A(x)}}\underset{\text{slab thickness}}{\underbrace{dx}}={\int }_a^b\left({\int }_c^{d}f(x,y)dy\right)dx.$

Inside-out padhein: pehle inner ${\int }_c^d\cdots dy$ ek slab ka face banata hai (Step 3), phir outer ${\int }_a^b\cdots dx$ us slab ko across sweep karta hai (Step 5). Yeh nested form iterated integral hai — parent note ka "FREEZE, FILL, FILE."

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Step 6 — DOOSRI taraf slice karo: hamaara choice forced nahi tha

KYA. Steps 3–5 dobara karo, lekin pehle $y$ freeze karo aur wall ko left-to-right ki jagah front-to-back slide karo.

KYO. Step 3 mein humne $x$ freeze karne ka choice kiya. Woh choice arbitrary thi — solid ko parwaah nahi ki hum chaaku kis taraf point karein. $y$ ko freeze karne se ek slice-area function of $y$ milta hai:

$B(y)={\int }_a^{b}f(x,y)dx,$

aur un slabs ko stack karne se wahi solid milta hai, isliye wahi volume.

PICTURE. Figure ek hi solid ke upar do chaaku directions side by side dikhata hai: Step-3 slice (grey, $y$ ke saath kaatti hai) aur naya slice (red, $x$ ke saath kaatta hai). Same loaf, do grains.

$V={\int }_c^d\underset{\text{fixed }y\text{ par face area}}{\underbrace{B(y)}}dy={\int }_c^d\left({\int }_a^{b}f(x,y)dx\right)dy.$

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Step 7 — Edge cases: yeh reasoning kahaan tootti hai?

KYA. Hum corners check karte hain: zero heights, negative heights, aur roofs jo itne wild hain ki slicing humse jhooth bolta hai.

KYO. Contract rule: reader ko kabhi koi aisa case nahi milna chahiye jo humne skip kiya. Teen matter karte hain.

PICTURE. Teen panels.

• Panel A — $f=0$ (degenerate). Flat floor, koi roof nahi: har slice-area $0$ hai, toh $V=0$. Sanity holds.
• Panel B — $f<0$ (floor ke neeche). Agar roof $xy$-plane ke neeche dip kare, toh woh slab negative volume count karta hai. Double integral isliye ek signed volume hai: floor ke upar ka space minus floor ke neeche ka space. Slicing verbatim kaam karti hai — areas bas ek sign carry karte hain.
• Panel C — $f$ not integrable. Agar $f$ blow up kare (unbounded, non-integrable), toh ek slice-area $A(x)$ shayad finite number bhi na ho, ya dono orders disagree kar sakti hain. Fubini ki guarantee khatam ho jaati hai. Isliye hypothesis "$f$ continuous / integrable on $R$" decoration nahi hai.

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Ek-picture summary

Sab kuch ek frame mein: solid (Step 1–2), ek red slice (Step 3), slice-area profile $A(x)$ (Step 4), slabs restacked (Step 5), aur twin knife direction (Step 6) — sab labelled taaki aap derivation seedha picture se padh sako.

Recall Feynman retelling — plain words mein poora walkthrough

Tumhare paas ek weird blob hai jo ek rectangular table par baitha hai, aur tum jaanna chahte ho usmein kitna stuff hai. Pehle tum guess karte ho table ko tiny squares se dhak ke aur har par ek chhota tower khada karke — towers add karo, roughly yahi volume hai, aur squares ko shrink karo toh exact ho jaata hai. Lekin zillions of towers add karna horrible hai. Toh iske bajaye tum ek chaaku pakadto ho. Ek direction freeze karo aur slice karo: har cut ek flat face dikhata hai, aur flat face ka ek ordinary area hota hai jo tum already jaante ho kaise measure karna hai. Ab chaaku ko along slide karo aur dekho face-area kaise grow aur shrink karta hai — woh ek simple humble curve $A(x)$ hai. Har slice ko thickness ka ek sliver do aur unhe glue karo: areas, add hoke, exact volume rebuild karte hain. Punchline: tum loaf ko doosre grain se bhi slice kar sakte the aur same loaf milta — toh same number. Woh "same loaf, either grain" hi Fubini hai. Sirf fine print: yeh ek real, tame loaf hona chahiye — agar roof kisi nasty way mein infinity pe shoot kare, toh slicing jhooth bol sakti hai, aur tumhe zyada careful rehna padega.

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Connections

• Double integrals over rectangles — Fubini's theorem — woh parent result jise yeh pictures prove karti hain.
• Riemann sums — Step 2 ka tower-sum ek 2-D Riemann sum hai.
• Volume by slicing (single-variable) — Steps 3–5 YEH HI idea hain, ek dimension upar lift kiya.
• Change of order of integration — Step 6 ki freedom use karta hai jab ek order hard ho.
• Fubini–Tonelli theorem — Step 7 Panel C ko absolute-value hypothesis se repair karta hai.
• Double integrals over general regions — flat floor ki jagah curvy floor leta hai.
• Triple integrals — ek 4-D "solid" ko teen directions mein slice karo.