Volume of revolution — disk method, washer method: Question bank

4.2.14 · D5 · Maths › Calculus II — Integration › Volume of revolution — disk method, washer method

Shuru karne se pehle, teen pictures apne dimaag mein rakho:

Ek disk ek solid coin hai — region axis of revolution ko bina kisi gap ke chhuta hai.
Ek washer ek coin hai jisme hole hai — region axis se door float karta hai, isliye ek ring banti hai.
Ek slice ki radius hamesha axis se curve tak ki distance hoti hai, kabhi bhi sirf " $y$ -value" nahi.

True or false — justify karo

TF1. "Disk aur washer do alag formulas hain jo alag se yaad karne padte hain."

False — disk sirf ek aisa washer hai jiska inner radius

r = 0

hai, isliye

π (R^{2} - 0^{2}) = π R^{2}

; actually ek hi idea hai: outer disk minus hole.

TF2. "Agar region axis of revolution ko chhuti hai, toh kabhi hole nahi banega."

True — hole (washer) tabhi banta hai jab region ki sabse paas wali edge axis se door hoti hai; axis ko chhune ka matlab hai ki near radius

0

hai.

TF3. "

y = x

aur

y = x^{2}

ke beech ki region ko x-axis ke baare mein rotate karne se wahi volume milta hai jitna y-axis ke baare mein rotate karne se."

False — axis badalne se radius (distance) aur slicing direction dono badte hain, isliye swept solids ki shape aur volume dono alag hoti hain.

TF4. "

π

se multiply karo toh

\int_{a}^{b} (R - r) d x

washer volume deta hai."

False — woh hoga

π \int (R - r) d x

, jo ek area-between-curves integrand ko volume problem mein mix karta hai; ring ki area

π R^{2} - π r^{2}

hai, na ki

π (R - r)

TF5. "Kyunki volume negative nahi ho sakta, radius hamesha positive honi chahiye, isliye absolute-value signs hata sakte hain."

Thoda sahi — radius ek distance hai isliye

\geq 0

hai, lekin hum safety ke liye

∣ f (x) - k ∣

likhte hain; khushkismati se hum isse square karte hain, aur squaring sign ko already mita deta hai, isliye answer automatically positive rehta hai.

TF6. "

y = 2

ke baare mein rotate karte waqt,

f (x)

ko radius ki tarah use kar sakte hain jab tak

f (x)

2

se neeche rehta hai."

False — radius axis tak ki distance

∣ f (x) - 2∣

hai, na ki

f (x)

;

f (x)

use karna

y = 0

se measure karna hai, jo galat line hai.

TF7. "Disk method ke liye axis ka x-axis ya y-axis hona zaroori hai."

False — tum kisi bhi horizontal line

y = k

ya vertical line

x = k

ke baare mein revolve kar sakte ho; bas har radius ko usi line se re-measure karo.

TF8. "Har slice ki thickness double karne se actual volume double ho jaati hai."

Conceptually False — thickness

d x \to 0

hai, ek infinitesimal jise hum sum karte hain, koi aisi quantity nahi jo scale ho sake; "

d x

double karna" sirf ek coarser approximation banata hai, koi alag solid nahi.

Error dhundho

SE1. "

[0, 1]

par

y = x

aur

y = x^{2}

ke beech ki region: volume

= π \int_{0}^{1} (x - x^{2})^{2} d x

Error — unhone difference ko square kiya; sahi hai

π \int_{0}^{1} (x^{2} - (x^{2})^{2}) d x

, matlab pehle har radius ko square karo phir subtract karo, kyunki

R^{2} - r^{2} \neq = (R - r)^{2}

SE2. "

[0, 4]

par

y = x

ko x-axis ke baare mein rotate karna:

V = π \int_{0}^{4} x d x

Error — radius

x

ko square karna zaroori hai:

π \int_{0}^{4} (x)^{2} d x = π \int_{0}^{4} x d x

; unhone

π r^{2}

mein

r^{2}

bhool gaye.

SE3. "Region ko x-axis ke baare mein rotate karte waqt, main

d y

thickness aur y-limits se slice karunga."

Error — slices axis ke ⊥ hote hain, isliye x-axis rotation mein

d x

thickness aur x-limits aate hain;

d y

y-axis rotation ka kaam hai.

SE4. "

1 \leq y \leq 3

wali region (ek horizontal strip) ko x-axis ke baare mein rotate karna ek radius

3

ki disk hai."

Error — strip x-axis ko nahi chhuti (sabse paas ki edge

y = 1

hai), isliye yeh ek washer hai:

R = 3

r = 1

, jo deta hai

π \int (3^{2} - 1^{2}) d x

SE5. "

y = 2

ke baare mein,

[0, 2]

par

y = x

ke neeche ki region: radius

= x - 2

, isliye

V = π \int_{0}^{2} (x - 2) d x

Do errors hain — radius ko square hona chahiye

(x - 2)^{2}

, aur isse forcefully positive banane ki zaroorat nahi kyunki squaring sign handle kar leti hai; sahi hai

π \int_{0}^{2} (x - 2)^{2} d x

SE6. "Washer ka outer radius hamesha top curve

f (x)

hoti hai."

Error — "outer" ka matlab axis se sabse door, na ki "sabse upar"; agar axis region ke upar ho, toh lower curve outer ho sakti hai, isliye distance se

R

identify karo, na ki kaunsa upar hai usse.

SE7. "Y-axis ke baare mein rotate karne ke liye main

y = f (x)

rakh kar sirf limits

c

aur

d

mein change karta hoon."

Error — tumhe

x = g (y)

mein invert karna hoga taaki radius slicing variable

y

mein express ho:

V = π \int_{c}^{d} [g (y)]^{2} d y

Why questions

WHY1. Ek patli slice ko cylinder kyun model karte hain, koi aur shape kyun nahi, jaise cone?

Infinitesimal thickness

d x

par curve barely change karta hai, isliye slice ki radius almost constant hoti hai — ek flat coin (cylinder), jiska

π r^{2} h

hum pehle se jaante hain; cone-jaisi taper limit

d x \to 0

mein khatam ho jaati hai.

WHY2. Har formula mein

π

integral ke bahar kyun aata hai?

Har slice ki area mein wahi constant factor

π

hota hai (from

π r^{2}

), aur constant factor ko sum se bahar nikala ja sakta hai — isliye integral se bhi bahar.

WHY3. Hum radius ko square kyun karte hain lekin thickness ki sirf first power kyun lete hain?

Circular face 2D hai isliye uski area

π r^{2}

radius ke square ke saath scale hoti hai; thickness ek remaining dimension hai jo ek akela factor

d x

deti hai — mile karke 3D volume banta hai.

WHY4. Definite integral, ek plain sum nahi, exact volume kyun deta hai?

Finite slabs sirf approximate karte hain; integral us Riemann sum ki limit hai jab

d x \to 0

, jo approximation error ko poori tarah hata deta hai.

WHY5. Ring ki area ke liye

R^{2} - r^{2}

sahi kyun hai lekin

(R - r)^{2}

galat kyun?

Ring outer disk minus inner disk hai,

π R^{2} - π r^{2}

;

(R - r)^{2} = R^{2} - 2 R r + r^{2}

expand karne par extra

- 2 R r + 2 r^{2}

terms dikh jaate hain jo wahan hone chahiye hi nahi — squaring linear nahi hai.

WHY6. Area Between Curves integrand

f - g

use karta hai jabki washers

f^{2} - g^{2}

kyun use karte hain?

Area heights (ek 1D difference

f - g

) sum karta hai; volume circular areas sum karta hai, aur har area radius ke square par depend karta hai, isliye areas ka difference

f^{2} - g^{2}

hota hai.

WHY7.

y = 0

ki jagah

y = k

ke baare mein rotate karte waqt radius ko re-measure kyun karna padta hai?

Radius ka matlab actual axis se distance hai; axis ko

y = k

par shift karna har distance ko

∣ f (x) - k ∣

shift kar deta hai, isliye

f (x)

rakhna galat line se measure karna hoga.

WHY8. u-substitution

π \int_{0}^{2} (2 - x)^{2} d x

jaisi shifted-axis integral ko kyun aasaan bana sakta hai?

u = 2 - x

set karne se shifted radius ek clean

π \int u^{2} d u

ban jaata hai, ek basic power integral,

(2 - x)^{2}

expand karne ka algebra chhupaate hue.

WHY9. Usi solid ke liye Shell Method kabhi kabhi disks/washers se zyada kyun prefer kiya jaata hai?

Shells axis ke parallel slice karte hain, isliye agar kisi region ko invert karna awkward ho (taaki

x = g (y)

mile) ya do alag washer integrals chahiye hon, toh natural variable mein ek shell integral zyada simple hota hai.

Edge cases

EC1. Agar region ki area zero ho (ek akela curve, koi enclosed region nahi) toh volume kya hoga?

Zero — koi width nahi toh sweep karne ke liye kuch nahi; "solid" sirf ek 2D surface of revolution hai, koi volume nahi.

EC2. Region axis ko exactly ek point par chhuti hai (jaise

y = x

y = 2

x = 2

par milna). Wahan disk hai ya washer?

Us point par inner radius

0

hai, isliye slice ek full disk hai; kuch bhi special nahi hota — washer smoothly us touch point par disk mein degenerate ho jaata hai.

EC3. Agar curve axis ke neeche jaati hai, matlab

[a, b]

ke kuch hisse par

f (x) < 0

ho, toh volume ka kya hoga?

Kuch bura nahi hota — radius distance

∣ f (x) ∣

hai, aur square karne se

[f (x)]^{2} \geq 0

milta hai sign se independent, isliye negative hissa bhi positive volume add karta hai (woh doosri taraf wahi coin sweep karta hai).

EC4. Agar do curves

[a, b]

ke andar cross karti hain, matlab "outer" aur "inner" swap ho jaate hain, toh kya ek washer integral use kar sakte hain?

Nahi — jahan swap hota hai,

R

aur

r

badal jaate hain; tumhe crossing point par alag integrals mein split karna hoga, har ek mein sahi

R

aur

r

ke saath.

EC5. Ek region ko ek aise line ke baare mein rotate karna jo uske interior se guzarti hai — kya yeh abhi bhi washer hai?

Ek washer nahi — axis region ko doosri taraf ke ek piece mein kaat deti hai; tum bade radius wale piece ko handle karo (ya dono ko sahi se add karo), kyunki ek slice ka outer radius uske inner radius se chhota nahi ho sakta.

EC6. Jab radius

f (x) = 0

poore interval par ho toh disk volume kya hoga?

Zero — har slice ki area

π \cdot 0^{2} = 0

hai; ek degenerate "solid" jo axis par collapse ho gaya hai.

EC7. Agar interval ki length zero ho (

a = b

), toh volume kya hoga?

Zero — integral

π \int_{a}^{a} [\dots] d x = 0

; axis ke saath koi extent nahi hai sweep karne ke liye, isliye koi solid nahi banta.

EC8. Ek aisi region ko rotate karna jo axis ke baare mein symmetric hai (upar aur neeche equal parts hain) — kya hum bottom half ko double karein ya ignore karein?

Ignore karein — top half wahi solid sweep karta hai jo bottom half karta hai (ghoomne par woh overlap karte hain), isliye sirf ek side use karo; double karna solid ko do baar count karna hoga.

Recall Ek-line survival kit

Radius = axis se distance (ise square karo). Thickness axis se match karti hai ( $d x$ x-axis ke liye, $d y$ y-axis ke liye). Ring = badi disk − chhoti disk, pehle squares, phir subtract.

Connections

Definite Integral as a Limit of Sums — slices ki limit exact kyun hai (WHY4).
Area Between Curves — $f - g$ vs $f^{2} - g^{2}$ ka distinction (WHY6).
Shell Method — alternative slicing jab inversion awkward ho (WHY9).
Volume by Cross-Sections — disks/washers circular special case hain.
u-substitution — shifted-axis integrals ke liye clean-up tool (WHY8).