Volume of revolution — disk method, washer method: Foundations

4.2.14 · D1 · Maths › Calculus II — Integration › Volume of revolution — disk method, washer method

Yeh page Volume of Revolution ka toolbox hai. Hum assume karte hain ki tumne kuch bhi nahi dekha. Har woh symbol jo parent note tumhe deta hai, yahan unpack kiya gaya hai, us order mein jo har ek ko pehle wale par lean karne deta hai.

0. Woh picture jis par hum baar baar laute hain

Figure — Volume of revolution — disk method, washer method

Figure ko dekho. Ek curve $y = f (x)$ ek horizontal line (axis) ke upar baitha hai. Hum unke beech ki shaded region leke, axis ko ek skewer ki tarah pakad kar, spin karte hain. Flat region blur hokar ek solid ban jaati hai — yahan, ek bowl/horn shape. Neeche sab kuch is picture ke parts ko precisely name karne ke baare mein hai.

1. $x$ aur $y$ — do directions

Simple words: $x$ batata hai ki tum kitna right ho; $y$ batata hai ki tum kitna upar ho.

Picture: do rulers ek corner par glued (yeh origin hai), ek flat pada, ek khada hua. Koi bhi point hai "right $x$ jao, phir upar $y$ jao".

Topic ko yeh kyun chahiye: ek region ko spin karne ke liye hume pehle usse locate karna hoga. Axis of revolution almost hamesha inheen rulers mein se ek hoti hai (ya unke parallel ek line).

2. $f (x)$ — ek function, yaani ek "height machine"

Simple words: $f (x)$ ek aisa rule hai jo ek number $x$ khaata hai aur ek height $y$ ugal deta hai. Letter $f$ sirf machine ka naam hai; $x$ woh hai jo tum use dete ho.

Picture: flat ruler par position $x$ par khade ho jao, seedha upar dekho, aur $f (x)$ utna hai ki curve tumhare bilkul upar kitni oonchi hai.

Topic ko yeh kyun chahiye: har slice ki radius exactly yahi height niklegi. Height machine nahi → radius nahi → volume nahi.

$g (x)$ sirf ek doosri height machine hai (alag letter taaki hum ek saath do curves ki baat kar sakein — washers ke liye zaroori).

3. Woh right triangle jo koi mention nahi karta — lekin radius ussi mein rehta hai

Simple words: axis se curve tak ka vertical segment hi ek slice ki radius hai. Uski length ek straight-line distance hai.

Picture: position $x$ par, point $(x, f (x))$ se axis tak ek vertical stick giraao. Jab axis x-axis hai toh stick ki length $f (x)$ hai (kyunki " $y = 0$ ke upar height" sirf $y$ -value hi hoti hai).

"Distance" kyun, sirf " $f (x)$ " kyun nahi? Jab hum baad mein ek shifted line $y = k$ ke around spin karte hain, toh stick ab curve se zero tak nahi jaati — woh curve se $k$ tak jaati hai. Uski length $∣ f (x) - k ∣$ ban jaati hai. Toh radius ki honest definition hai axis se curve tak ki distance, aur $f (x)$ sirf special case $k = 0$ hai.

4. $π$ aur ek circle ka area

Simple words: $π \approx 3.14159$ woh fixed number hai jo circle ki radius ko uske area aur circumference se jodhta hai. Yeh kabhi nahi badlta.

Picture: koi bhi circle lo, use thin pizza slices mein kaato, unhe ek almost-rectangle mein re-arrange karo jiska height $r$ aur width $π r$ hai — rectangle ka area $π r^{2}$ hi circle ka area hai.

Topic ko yeh kyun chahiye: har slice ka flat face ek circle hota hai (yahi spinning produce karti hai). Uska area slice-volume formula ka agla hissa hai.

5. Cylinder — kyun ek thin slice ka volume jaana jaata hai

Simple words: cylinder ek circle hai jise thodi door sideways kheeнcha gaya hai. Uska volume circle ka area hai times kitni door kheeнcha gaya.

Picture: ek coin. Uska face $π r^{2}$ area ka ek circle hai; uski thickness $h$ hai. "Stuff ki miqdar" = face area $\times$ thickness.

Topic ko yeh kyun chahiye: solid ka ek asli slice thodi curved sides rakhta hai, toh uska volume mushkil hai. Lekin agar slice ultra-thin hai, toh curve usके across barely change hoti hai, toh slice almost ek perfect cylinder hai — jiska volume hume pata hai. Yahi trick hai.

6. $Δ x$ , phir $d x$ — "width ka ek tiny bit"

Simple words: $Δ x$ (padho "delta x") matlab hai " $x$ mein ek chhota sa change" — ek slice ki thickness. $d x$ matlab wahi cheez hai, lekin zero ki taraf shrunk — ek infinitely thin slice.

Picture: upar figure mein slab ki width. Use thinner aur thinner karo; $Δ x \to d x$ .

Topic ko yeh kyun chahiye: cylinder formula mein thickness $h$ yahi width hai. $Δ V \approx π [f (x)]^{2} Δ x$ likhna kehta hai "is slice ka volume ≈ face area × uski width". Jaise slices infinitely thin hoti hain approximation exact ho jaata hai.

7. $\sum$ aur $\int$ — saari slices add karna

Simple words: $\sum$ (capital sigma) matlab hai "inhe add karo". $\int$ (lamba S) matlab hai "infinitely many infinitely-thin pieces add karo" — ek smooth, exact total.

Picture: finitely many slabs (ek chunky staircase estimate) morphing, jaise woh thinner hoti hain, ek smooth filled solid mein. Staircase error gayab ho jaata hai.

Topic ko yeh kyun chahiye: ek slice deta hai $π r^{2} d x$ . Poora solid $x = a$ se $x = b$ tak saari slices ka sum hai. $Δ x$ ke saath finite sum = ek rough guess; $Δ x \to 0$ ki limit $\sum$ ko $\int$ mein badal deti hai aur guess sach mein. Yahi exactly Definite Integral as a Limit of Sums machinery hai.

$a$ aur $b$ limits of integration hain — jahan region axis ke along shuru aur khatam hoti hai. Yeh region ke sabse chhote aur sabse bade $x$ hain.

8. Notation cheat-sheet (har mark ka matlab)

9. Do curves → washer ko $R$ aur $r$ chahiye

Simple words: agar region axis ke upar float karta hai neeche gap ke saath, toh har spun slice ek ring hai, solid coin nahi — gap hole ban jaata hai.

Picture: region ek upper curve $f (x)$ aur ek lower curve $g (x)$ ke beech trapped hai. Use spin karo: upper curve outer edge trace karti hai (radius $R = f (x)$ ), lower curve hole ki inner edge trace karti hai (radius $r = g (x)$ ).

Topic ko yeh kyun chahiye: yeh washer method mein sirf ek nayi idea hai — kaun si curve door hai ( $R$ ) aur kaun si paas ( $r$ ). Area $π R^{2} - π r^{2}$ hai, jo seedha integral mein jaata hai. Area Between Curves se compare karo, jahan tum heights $f - g$ subtract karte (squaring nahi) — alag sawaal, alag integrand.

Prerequisite map

Equipment checklist

Khud test karo — right side cover karo.

Main

f (x)

ka matlab diye gaye

x

ke liye padh sakta/sakti hoon

Haan — yeh curve ki height directly position

x

ke upar hai.

Main radius

r

wale circle ka area jaanta/jaanti hoon

π r^{2}

Main ek cylinder (coin) ka volume jaanta/jaanti hoon

face ka area

\times

thickness

= π r^{2} h

Main bata sakta/sakti hoon ek thin slice ki radius kya hai, x-axis ke around spin karte waqt

curve ki height,

r = f (x)

Main bata sakta/sakti hoon line

y = k

ke around spin karte waqt radius

distance

∣ f (x) - k ∣

Main jaanta/jaanti hoon kyun hum ek slice ko cylinder approximate karte hain

ek ultra-thin slice barely curve karti hai, toh yeh almost ek perfect coin hai jiska volume hume pata hai.

Main jaanta/jaanti hoon

Δ x

integral ke andar kya ban jaata hai

d x

— ek infinitely thin width; sum exact ban jaata hai.

Main jaanta/jaanti hoon

\int_{a}^{b}

kya karta hai

x = a

x = b

tak saari slices ko ek exact total mein add karta hai.

Main jaanta/jaanti hoon ek ring ka area

π R^{2} - π r^{2}

hai,

π (R - r)^{2}

nahi

Haan — do areas subtract karo; difference ka square karna galat hai.

Main ek disk aur washer mein fark bata sakta/sakti hoon

disk = koi hole nahi (region axis ko touch karti hai); washer = hole hai (region mein axis se gap hai).

Connections

Definite Integral as a Limit of Sums — $\sum \to \int$ bridge yahan rehta hai.
Area Between Curves — same two-curve setup, lekin tum heights subtract karte ho, squared radii nahi.
Volume by Cross-Sections — disks/washers circular special case hain.
Shell Method — alternative jo axis ke parallel slice karta hai.
u-substitution — shifted-axis integrals evaluate karne ka algebra tool.