Double integrals over general regions — Type I and II: Foundations

4.4.17 · D1 · Maths › Multivariable Calculus › Double integrals over general regions — Type I and II

Parent page par koi bhi formula padhne se pehle, tumhe har symbol ko bina hesitation ke apna banana hoga. Yeh page unhe sab ground up se build karta hai, har ek apni jagah earn karta hai agle ke aane se pehle.

1. Plane aur ek point $(x, y)$

Figure — Double integrals over general regions — Type I and II

$x$ ::= kitna daayein (ya baayein, agar negative ho).
$y$ ::= kitna upar (ya neeche, agar negative ho).
Topic ko yeh kyun chahiye: har region $D$ aise points ka set hai, aur $f (x, y)$ har ek par ek value padhta hai. Point nahi, toh integral nahi.

2. Ek region $D$ aur symbol $\subseteq$

Parent page jo curly-brace notation use karta hai, $D = {(x, y) : a \leq x \leq b, g_{1} (x) \leq y \leq g_{2} (x)},$ yeh aise zor se padha jaata hai: " $D$ un saare points $(x, y)$ ka set hai jaise ki ( $:$ matlab such that) $x$ , $a$ aur $b$ ke beech mein hai, aur $y$ do curves ke beech mein hai." Braces ${}$ ka matlab hai "ka collection," aur colon $:$ woh gate hai jo entry rules list karta hai.

Topic ko yeh kyun chahiye: poora method $D$ ko exactly isi "such-that" form mein likhne ke baare mein hai. Ek baar kar lo, limits apne aap aa jaate hain.

3. Ek function $f (x, y)$ aur surface $z = f (x, y)$

Topic ko yeh kyun chahiye: $f$ woh cheez hai jo integrate ho rahi hai; agar $f \equiv 1$ (constant height 1) ho toh trapped "hawa" sirf $D$ ka area hai — parent page par sanity check.

4. Integral sign $\int$ — ek infinite sum

Parts padho:

$\int$ ::= "infinitely many tiny pieces add karo."
$a$ (bottom) aur $b$ (top) ::= jahan sweep shuru aur khatam hoti hai.
$d x$ ::= ek slice ki infinitely thin width, aur yeh us variable ka naam hai jise hum sweep kar rahe hain.

Topic ko yeh kyun chahiye: parent ka iterated integral sirf do aise sweeps hain ek doosre ke upar rakhke — tum $\int_{a}^{b} \int_{g_{1}}^{g_{2}}$ nahi padh sakte jab tak ek ko apna nahi bana lete.

5. Ek $\int$ se do tak — iterated integral

Golden reading rule, ab jab tumhare paas symbols hain:

Inner limits ( $g_{1} (x), g_{2} (x)$ ) outer variable $x$ par depend kar sakte hain — kyunki jab strip sideways slide hoti hai, uska top aur bottom move karta hai.
Outer limits ( $a, b$ ) plain constants hone chahiye — kyunki har strip ko sum karne ke baad kuch bachta nahi depend karne ke liye.
Topic ko yeh kyun chahiye: yahi Type I / Type II ka poora deliverable hai — $\iint_{D}$ ko do ordinary sweeps mein turn karna.

6. $d A$ — area element

Topic ko yeh kyun chahiye: $d A$ coordinate-free hai (isse koi fark nahi padta tum kis taraf se slice karo). $d A = d y d x$ vs $d x d y$ choose karna exactly Type I aur Type II ke beech ka choice hai.

7. Curves as boundaries: $g_{1} (x), g_{2} (x)$ aur $h_{1} (y), h_{2} (y)$

8. Prerequisite map

Upar se neeche padho: points ek region banate hain aur ek function ko feed karte hain; function ek surface banata hai; single sweeps iterated sweeps mein combine hote hain; boundary curves aur area element limits supply karte hain — aur yeh sab Type I / Type II formulas mein pour ho jaata hai.

9. Yeh prerequisites aage kahan le jaate hain

Jab upar ke symbols second nature ban jaayein, yeh vault pages readable ho jaate hain:

Fubini's Theorem — guarantee ki inner-then-outer, outer-then-inner ke barabar hai.
Double integrals over rectangles — constant-limit warm-up case.
Changing the order of integration — boundary ko re-read karke $d y d x$ ko $d x d y$ se swap karna.
Area between two curves — $f \equiv 1$ special case disguise mein.
Polar coordinates double integrals, Centre of mass and moments, Jacobian and change of variables — baad ke payoffs jo sab yahi machinery reuse karte hain.

Equipment checklist

Khud test karo — right side cover karo aur zor se jawaab do.

$(x, y)$ kya locate karta hai? ::: Ek single point: $x$ daayein, $y$ upar, flat plane $R^{2}$ par.
$D = {(x, y) : \dots}$ ko words mein padho. ::: "Un saare points $(x, y)$ ka set jaise ki listed conditions hold karti hain."
$\subseteq$ ka matlab kya hai? ::: "Is a subset of" — left set ka har point right set mein bhi hai.
$f (x, y)$ kya hai? ::: Ek machine jo ek point ko ek number mein turn karti hai (plane ke upar ek height).
$z = f (x, y)$ kaun sa surface draw karta hai? ::: Region ke upar hover karti ek curved roof / tent.
Symbol $\int$ kya represent karta hai? ::: Infinitely many infinitely-thin pieces ka ek continuous sum.
$\int$ par bottom aur top numbers ka matlab kya hai? ::: Jahan sweep shuru aur khatam hoti hai.
$d x$ ke do kaam kya hain? ::: Tiny slice width mark karta hai aur us variable ka naam deta hai jo sweep ho raha hai.
Iterated integral mein, kaun se limits mein variable ho sakta hai? ::: Sirf inner limits mein, aur sirf outer variable — outer limits mein kabhi nahi.
Outer limits constants kyun hone chahiye? ::: Saari strips sum karne ke baad kuch nahi bachta depend karne ke liye, isliye result ek plain number hai.
$d A$ kya hai, aur yeh $d y d x$ kaise banta hai? ::: Area ka ek tiny tile; concretely ek $d x$ -wide, $d y$ -tall rectangle.
$g_{1} (x)$ vs $h_{1} (y)$ — kya fark hai? ::: $g$ , $x$ se $y$ deta hai (vertical strips, Type I); $h$ , $y$ se $x$ deta hai (horizontal strips, Type II).