Foundations — Double integrals over rectangles — Fubini's theorem

4.4.16 · D1 · Maths › Multivariable Calculus › Double integrals over rectangles — Fubini's theorem

Tumhare parent note Double integrals over rectangles — Fubini's theorem ko padhne se pehle, tumhe har woh symbol samajhna hai jo woh silently assume karta hai. Yeh page unhe ek-ek karke, usi order mein build karta hai jisme woh stack hote hain. Yahan koi bhi symbol use nahi hoga jab tak woh pehle explain na ho.

1. Points aur coordinates: $(x,y)$ aur plane

Topic ko yeh kyun chahiye. Jis floor par hum integrate karte hain woh isi plane mein hoti hai. Parent ke Riemann sum ka har "sample point" $(x_i^{\ast },y_j^{\ast })$ yahan sirf ek dot hai. Agar $(x,y)$ bilkul clear nahi hai, toh poora sum dhundla ho jaata hai.

2. Do variables ka function: $z=f(x,y)$

Ek floor imagine karo jiske upar ek kapda tana hua hai. Har floor-point ke upar kapda kisi height par hota hai — woh height $f(x,y)$ hai. Jaise-jaise tum $(x,y)$ ko slide karte ho, kapda upar-neeche hota hai: yeh ek surface hai.

Topic ko yeh kyun chahiye. Parent ka phrase "surface $z=f(x,y)$" aur "volume under the surface" tabhi samajh aata hai jab tum $f$ ko har floor-point ke upar roof height ke roop mein dekho. Agar $f(x,y)$ negative hai, toh roof floor ke neeche jaati hai — baad mein signed volume ke liye yeh yaad rakhna.

3. Rectangle $R=[a,b]\times [c,d]$

Set language mein parent likhta hai $R=[a,b]\times [c,d]=\{(x,y):a\le x\le b,c\le y\le d\}.$

Isko zor se padho: "points $(x,y)$ ka set aisa ki $x$, $a$ aur $b$ ke beech ho AUR $y$, $c$ aur $d$ ke beech ho." Colon $:$ ka matlab "such that" hai; comma ka matlab "and" hai.

Topic ko yeh kyun chahiye. $R$ woh floor hai jiske upar hum volume banate hain. Kyunki iske edges fixed numbers $a,b,c,d$ hain (curved nahi, ek doosre par depend nahi karte), integration limits plain constants hongi — aur yahi reason hai ki rectangular Fubini theorem easy first case hai, Double integrals over general regions se pehle.

4. Todna: $\Delta x$, $\Delta y$, $\Delta A$, aur partition $P$

$[a,b]$ ko $\Delta x$ width ki patli strips mein kaato aur $[c,d]$ ko $\Delta y$ height ki patli strips mein. Grid lines $R$ ko choti sub-rectangles mein kaatti hain. Is poore grid ko partition $P$ kehte hain.

Topic ko yeh kyun chahiye. $\Delta A$ base aur $f(x_i^{\ast },y_j^{\ast })$ height wali ek tower ka volume $f\cdot \Delta A$ hai. $\Vert P\Vert$ ko shrink karna towers ki staircase ko smooth true volume mein badalta hai. Yahi definition ki dhadkan hai — aur yeh Riemann sums ka 2-D twin hai.

5. Sample point $(x_i^{\ast },y_j^{\ast })$ aur star $\ast$

Topic ko yeh kyun chahiye. Hum poore tile ki heights use nahi kar sakte, toh ek height $f(x_i^{\ast },y_j^{\ast })$ choose karte hain jo us tile par tower ko represent kare. Jab tiles dots mein shrink ho jaati hain, choice matter karna band ho jaati hai — ek dot-sized tile mein har point ki height lagbhag same hoti hai.

6. Summation signs $\sum _i\sum _j$

Toh parent ka Riemann sum $S={\sum }_i{\sum }_{j}f(x_i^{\ast },y_j^{\ast })\Delta A$ padha jaata hai: "har tile par har choti tower ka volume add karo."

Topic ko yeh kyun chahiye. Yeh finite sum approximation hai. Integral uska limit hai. Notice karo ki double sum pehle se hi Fubini hint karta hai: across-then-up add karna equals up-then-across add karna — do sigmas ka order total nahi badalta.

7. Integral signs: $\int$, $\iint$, aur $dA$

8. Iterated integral: ${\int }_a^b{\int }_c^{d}fdydx$

Topic ko yeh kyun chahiye. 2-D limit directly compute karna brutal hai. Iterated integral ek mushkil 2-D problem ko do easy 1-D integrals mein badal deta hai jo tum pehle se karna jaante ho. Yahi Fubini ka poora practical payoff hai — aur jab order swap karna seekh lo, tum Change of order of integration unlock kar lete ho.

9. Prerequisite map

Isko aise padho: points aur functions ek rectangle par ek surface banate hain; rectangle ko kaatna Riemann sum banata hai; uska limit double integral hai; slicing use iterated integrals mein badal deta hai; dono slicing directions ko match karna hi Fubini hai.

Equipment checklist

Reveal karne se pehle har jawab zor se bolo.

Connections

• Riemann sums — yahan double sum uska 2-D version hai.
• Volume by slicing (single-variable) — jahan se $A(x)$ aur $\int A(x)dx$ aate hain.
• Double integrals over rectangles — Fubini's theorem — parent note jiske liye yeh page tumhein prepare karta hai.
• Double integrals over general regions — aage, jab $R$ ke edges curve ho jaate hain.
• Change of order of integration — Fubini par trust karne ke baad $dxdy$ swap karna.
• Fubini–Tonelli theorem — "swapping kab allowed hai" ka deeper statement.
• Triple integrals — ek aur dimension, tiles ki jagah tiny boxes.