Volume of revolution — shell method: Question bank

4.2.15 · D5 · Maths › Calculus II — Integration › Volume of revolution — shell method

True or false — justify

TF1. "Shell method hamesha

d x

thickness use karta hai."

False. Thickness strip se match hoti hai, aur strip axis ke parallel chalni chahiye. Horizontal axis ke baare mein strips horizontal hoti hain, toh thickness

d y

hoti hai.

TF2. "Kisi region ko door wale axis ke baare mein rotate karne par hamesha bada volume milta hai."

False. Radius door wali side par barhti hai, lekin height profile aur region kahan baith raha hai — yeh bhi matter karte hain. Parent ka apna example

x = 0

aur

x = 3

dono ke baare mein

8 π

deta hai — shift ke bawajood same volume.

TF3. "Shell integral mein radius hamesha integration variable (

x

y

) hoti hai."

False. Radius strip se axis ki distance hoti hai. Yeh

x

ke barabar tab hi hoti hai jab axis

y

-axis ho

x = 0

par;

x = c

ke baare mein yeh

∣ x - c ∣

hoti hai.

TF4. "Do curves

y = f (x)

aur

y = g (x)

ke beech ke region ke liye, shell height

f (x) + g (x)

hoti hai."

False. Height top minus bottom hoti hai, yaani

f (x) - g (x)

(strip ki actual length). Add karne se double-count hoga, jo nonsense dega.

TF5. "Disk aur shell methods ko freely swap kar sakte hain; dono same number dete hain."

Value ke liye True, effort ke liye false. Dono same volume compute karte hain, lekin Volume of revolution — disk and washer method dekho — ek method ko aksar ek ugly inverse ki zaroorat padti hai jo doosra avoid karta hai. Aasaan wala chuno.

TF6. "Agar rotation ka axis region se hokar guzarta hai, toh shell setup unchanged rehti hai."

False. Radius

∣ x - c ∣

axis ke aas-paas

(x - c)

ka sign change karti hai, isliye tumhe

x = c

par integral split karni padti hai aur har side par absolute distance use karni hoti hai. Warna negative-radius terms galat tarike se real volume cancel kar dete hain.

TF7. "Jis shell ki height zero hai, woh zero volume contribute karti hai."

True.

d V = 2 π r h d x

; agar

h = 0

hai toh strip ki koi length nahi, toh tube empty hai. Yahi wajah hai ki jahan curves milti hain woh endpoints kuch contribute nahi karte.

TF8. "

2 π

factor circle ki area se aata hai."

False. Yeh circumference

2 π r

se aata hai — woh length jo tube ko cheer ke flat sheet mein unroll karne par milti hai. Disks area

π r^{2}

use karte hain; shells perimeter use karte hain.

TF9. "Shells aur disks ek hi problem ke liye same variable par integrate karte hain."

False. Ye zyaatar opposite variables use karte hain: same axis ke liye, disk strips perpendicular hoti hain (ek variable) jabki shell strips parallel hoti hain (doosra). Yahi exact reason hai kyun ek

f

ko invert karne se bachta hai.

Spot the error

SE1. Ek student

y = x^{2}

0 \leq x \leq 2

x = 3

ke baare mein rotate karta hai aur likhta hai

V = \int_{0}^{2} 2 π x \cdot x^{2} d x

Radius galat hai:

x = 3

ke baare mein yeh

∣ x - 3∣ = 3 - x

hai,

x

nahi. Usne aadat se

y

-axis wali radius use kar li; correct hai

\int_{0}^{2} 2 π (3 - x) x^{2} d x

SE2.

x

-axis ke baare mein rotate karte hue, ek student likhta hai

V = \int_{a}^{b} 2 π x f (x) d x

with

d x

Axis horizontal hai, isliye strips horizontal honi chahiye thickness

d y

ke saath; radius

∣ y ∣

hai aur height

x_{right} - x_{left}

hai. Horizontal axis ke saath

d x

aur

x

-radius milana classic mismatch hai.

SE3.

[0, 1]

par

y = x

aur

y = x^{2}

ke beech ke region ke liye

y

-axis ke baare mein, ek student

h = x^{2} - x

set karta hai.

Sign ulta hai.

(0, 1)

par,

x > x^{2}

, isliye top

y = x

hai: height

x - x^{2}

honi chahiye. Jaise likha hai,

h

negative hai aur volume negative aata hai.

SE4. Ek student likhta hai

d V = π r h d x

, "disk

π

use karta hai isliye."

Factor of

2

missing hai. Tube unroll karne par width

2 π r

ki ek sheet milti hai (ek poora loop), isliye

d V = 2 π r h d x

. Disk ka

π

adha yaad rakhna yahan trap hai.

SE5. Thickness

d x

li gayi hai lekin height

h = y = x

likhi gayi hai... phir student "integrates in

y

Variable inconsistency. Agar thickness

d x

hai, toh radius, height, aur limits sab

x

ke functions hone chahiye.

d x

element ko

y

ke respect mein integrate nahi kar sakte.

SE6.

x = 3

ke baare mein

0 \leq x \leq 2

par region ke saath rotate karte hue, ek student

r = x - 3

likhta hai (absolute value nahi).

x - 3

poore region par negative hai, toh yeh radius negative hai. Correct radius distance

3 - x > 0

hai. Yahan sign error poore volume ko negative kar deti.

SE7. Ek student shell integral (

y

-axis ke baare mein) ki limits

y

-values se set karta hai kyunki "region

y = 0

y = 4

tak jaata hai."

Limits thickness variable se match honi chahiye. Thickness

d x

hai, isliye limits strips ki

x

-range hain,

y

-range nahi.

Why questions

WHY1.

π h (2 r d x + d x^{2})

mein

d x^{2}

term kyun drop ho jaata hai?

Yeh second order hai: jab

d x \to 0

d x^{2}

d x

se kahin zyada tezi se chhotha hota hai, isliye total sum mein iska contribution limit mein vanish ho jaata hai. Dekho Definite integral as a limit of Riemann sums.

WHY2.

y

-axis ke baare mein

y = f (x)

ke liye shell method aksar disks se better kyun hoti hai?

Shells natural variable $x$ rakhte hain, isliye tumhe kabhi

y = f (x)

x = f^{- 1} (y)

ke liye solve nahi karna padta, jo ugly ya impossible ho sakta hai.

WHY3. Shell strips axis ke parallel kyun chalni chahiye, perpendicular nahi?

Axis ke parallel ek strip rotate hone par ek tube (shell) sweep karti hai; ek perpendicular strip ek disk/washer sweep karti hai. Swept solid ki geometry method decide karti hai.

WHY4. Shell height "top − bottom" area between curves se same idea kyun hai?

Strip ki length exactly do curves ke beech vertical gap hai — wahi

y_{top} - y_{bot}

integrand Area between two curves se. Shells bas har strip ko

2 π r

se weight karte hain aur use revolve karte hain.

WHY5. Circumference

2 π r

kyun hai,

π r

π d /2

kyun nahi?

Radius

r

wale circle ke poore ek chakkar ki length

2 π r

hoti hai — yeh

π

ki definition se hai. Shell unroll karne par woh poora loop flat sheet ki width ki tarah lay out hota hai.

WHY6. Same volume

x

mein integrate karne se (shells) ya

y

mein (disks) kyun aa sakta hai?

Dono same solid ko patli pieces mein partition karte hain; total is baat se independent hai ki tum kaise slice karte ho. Sirf algebra alag hai — yeh Choosing dx vs dy strips mein "

d x

d y

" choice hai.

WHY7. Jab axis region se guzarti hai toh integral split kyun karni padti hai?

Distance

∣ x - c ∣

mein

x = c

par ek kink hai (ek side par

c - x

hai, doosre par

x - c

). Absolute value wale integrand ko us point par split karke handle kiya jaata hai taaki har piece smooth aur positive ho.

Edge cases

EC1. Jab region ki width zero ho (maan lo

a = b

) toh volume kya hoga?

Zero. Integral

\int_{a}^{a} = 0

; revolve karne ke liye koi strip nahi, toh koi solid nahi banta.

EC2. Jis shell ka radius

r = 0

ho (strip exactly axis par baith rahi ho) uska kya hoga?

Iska circumference

2 π r = 0

hoga, isliye

d V = 0

. Ek degenerate tube jo ek line mein collapse ho gayi ho uska koi volume nahi — yeh akela strip kuch contribute nahi karta.

EC3. Agar ek region

y

-axis ke dono taraf hai (positive aur negative

x

dono hain) aur hum

y

-axis ke baare mein rotate karte hain, toh radius kya hai?

r = ∣ x ∣

, aur zyaatar tum $x = 0$ par split karte ho kyunki dono sides swept solid mein overlap karti hain. Sirf

r = x

use karne par left side par negative contributions aayenge.

EC4. Ek region axis

x = c

ke left mein poori tarah hai (toh

x < c

). Radius kya hai, aur kuch alag hai?

r = c - x > 0

, wahi positive distance. Setup "right of axis" case se identical hai — sirf absolute value ke andar sign flip hota hai, aur

∣ x - c ∣

ise automatically handle karta hai.

EC5. Ek region ko horizontal line

y = c

ke baare mein rotate karna jo poore region ke neeche hai — radius kaise set hogi?

Har strip ke liye

r = y - c > 0

, horizontal strips ke saath thickness

d y

aur height

x_{right} - x_{left}

. Koi splitting nahi chahiye kyunki region kabhi

y = c

cross nahi karta.

EC6. Curves integration endpoints par milti hain (jaise

y = x

aur

y = x^{2}

x = 0, 1

par milti hain). Wahan height kya karti hai?

Height dono meeting points par

\to 0

hoti hai, isliye boundary shells vanish ho jaate hain. Solid ends par smoothly kuch nahi reh jaata — koi discontinuity nahi, koi leftover cap nahi.

EC7. Kya shell height kabhi horizontal length ho sakti hai jabki thickness

d x

ho?

Nahi. Thickness

d x

ke saath strips vertical hain, isliye unki length vertical hai (ek

y

-difference). Horizontal heights (

x_{R} - x_{L}

) thickness

d y

ke saath jaate hain. Inhe match karna orientation rule hai.

Recall Ek-line self-test

Upar har answer cover karo aur reason dobara derive karo, sirf verdict nahi. Agar tum explain kar sakte ho kyun ek false statement false hai, tabhi concept tumhara hai.

Connections

4.2.15 · Shell Method — parent derivation jise yeh traps guard karte hain.
Choosing dx vs dy strips — orientation decision jo in half misconceptions ke peeche hai.
Volume of revolution — disk and washer method — "same value, different effort" comparison.
Area between two curves — jahan se "top − bottom" aata hai.
Definite integral as a limit of Riemann sums — kyun $d x^{2}$ term vanish hota hai.