Surface area of revolution: Question bank — Hinglish deep dive

4.2.17 · D5 · Maths › Calculus II — Integration › Surface area of revolution

In questions mein use hone wale vocabulary ke quick reminders (sab parent note mein build kiye gaye hain):

$d s$ = arc-length element, curve ke ek tiny slanted piece ki length, $d s = 1 + (y^{'})^{2} d x$ .
$d x$ = us piece ki horizontal shadow (axis par uski projection).
Radius = curve ke kisi point ki rotation axis se doori (woh kitna travel karta hai ek full spin mein, $2 π$ se divide karke).
Frustum = ek cone jiska tip kaat diya gaya ho; har curve-piece ek thin band sweep karta hai.

True or false — justify karo

TF1. X-axis ke baare mein ek curve rotate karne par, surface area integrand

2 π y d x

hota hai.

False. Yeh

2 π y d s

hota hai;

d x

use karne se har piece ki sirf horizontal shadow li jaati hai aur curve ke har us hisse ko undercount kiya jaata hai jo flat nahi hai.

TF2. Agar ek curve

[a, b]

par bilkul horizontal hai (constant

y = c

), to wahan

d s = d x

hoga.

True. Ek flat piece mein

d y = 0

hota hai, isliye

d s = 1 + 0 d x = d x

— slant, shadow ke barabar tabhi hoti hai jab kuch upar ya neeche na jaye.

TF3. Surface area of revolution mein

π r^{2}

zaroor kahin use hota hai, bilkul volume of revolution ki tarah.

False. Volume mein disks (

π r^{2}

) stack hoti hain; surface mein rings wrap hoti hain jinका contribution circumference

2 π r

times slant

d s

hota hai. Contrast ke liye Volume of revolution (disk & shell) dekho.

TF4. Segment

y = x

x \in [0, 1]

ko x-axis ke baare mein rotate karne par wahi surface area milti hai jo y-axis ke baare mein rotate karne par milti hai.

Is particular line mein

y = x

ke baare mein symmetry ki wajah se True hai, lekin generally nahi — radius factor

y

x

mein badal jaata hai, isliye sirf symmetric configurations mein hi coincide hota hai.

TF5. Agar ek curve ki har length double kar do (scale by 2), to surface of revolution ki area bhi double ho jaati hai.

False. Area, length ke square ke saath scale hoti hai, isliye yeh

4 \times

badi ho jaati hai — radius aur

d s

dono double ho jaate hain.

TF6. Factor

1 + (y^{'})^{2}

kabhi bhi 1 se kam nahi ho sakta.

True. Yeh

1

sirf tab hota hai jab

y^{'} = 0

aur kisi bhi nonzero slope ke liye badhta hai, kyunki ek square kabhi negative nahi hota; slant edge hamesha apni shadow se kam se kam utni lambi hoti hai.

TF7.

y = f (x)

ko line

y = 3

ke baare mein rotate karne par bhi radius

y

rehti hai.

False. Radius axis se doori hoti hai, yahan

∣ y - 3∣

;

y

use karna silently assume karta hai ki axis

y = 0

hai.

TF8. Ek aise curve ke liye jo x-axis ke neeche jaati hai,

2 π y

use karna chahiye jahan

y

possibly negative ho.

False. Doori kabhi negative nahi hoti, isliye radius

2 π ∣ y ∣

hai; negative radius nonsensical negative area contributions deta.

Error dhundo

SE1. "X-axis ke baare mein

x = g (y)

rotate karne ke liye,

S = \int 2 π y 1 + (d x / d y)^{2} d y

Radius galat hai: y-axis se doori

x = g (y)

hai,

y

nahi. Yeh

2 π x = 2 π g (y)

hona chahiye;

y

height measure karta hai, axis se doori nahi.

SE2. "

d s = 1 + (y^{'})^{2} d x

hai, isliye main

\int 2 π y 1 + (y^{'})^{2} d y

y

-limits par integrate karunga."

Variable mismatch hai:

d s

d x

form,

x

-limits aur

d x

ke saath pairing maangta hai. Agar

y

-limits par switch karo to

d s = 1 + (d x / d y)^{2} d y

likhna hoga.

SE3. "Slant

L

, radius

r

wale ek full cone ki lateral area

π r^{2} L

hai."

Area

π r L

hai (cone ko unroll karke circular sector banao).

π r^{2}

flat base ka disk area hoga, jo bilkul alag cheez hai — Frustum and cone geometry dekho.

SE4. "Frustum lateral area

2 π r_{1} ℓ

hai jahan

r_{1}

chhota radius hai."

Average radius use karni chahiye:

π (r_{1} + r_{2}) ℓ

. Sirf chhota radius use karne se flaring band undercount hoti hai; mean circumference hi wrap hoti hai.

SE5. "

y = r^{2} - x^{2}

ko x-axis ke baare mein rotate karne par, main

0

r

tak integrate karunga aur yahi poora sphere hai."

Yeh sirf hemisphere cover karta hai. Poore sphere ke liye

x \in [- r, r]

chahiye; symmetry se

[0, r]

result double kar sakte ho, lekin sirf limit relabel karke nahi.

SE6. "Pappus ke theorem se,

S = 2 π \overset{r}{ˉ} L

jahan

\overset{r}{ˉ}

region ka centroid hai."

Surface area ke liye

\overset{r}{ˉ}

curve (arc) ka centroid hai, enclosed area ka nahi. Dono ko mix karna surface theorem ko volume version ke saath confuse karta hai — Pappus's theorem dekho.

SE7. "Curve ek point par axis ko cross karti hai, isliye surface mein wahan ek hole hai."

Jahan curve axis se milti hai wahan radius

0

hoti hai, isliye woh ring ek point par shrink ho jaati hai — surface smoothly ek tip par close ho jaati hai, koi hole nahi. Sphere ke poles ke baare mein socho.

Why questions

W1. "Kitna lamba" factor mein flat

d x

ki jagah slant length

d s

kyun sahi hai?

Surface tilted curve ke saath hi wrap hoti hai; ek steep piece apni short horizontal shadow se zyada surface cover karti hai, aur sirf

d s

hi woh sahi tilted length measure karta hai.

W2. Frustum formula mein dono radii ka average kyun use hota hai?

Band ko unroll karne par ek aisi shape milti hai jiska width linearly

2 π r_{1}

2 π r_{2}

tak vary karta hai, isliye uski area mean circumference

2 π \frac{r _{1} + r _{2}}{2}

times slant hai — bilkul trapezoid ke dono ends average karne ki tarah.

W3. Jab band infinitesimal ho jaata hai to

r_{1} \approx r_{2} \approx y

kyun hota hai?

Ek vanishingly short piece par curve ki height barely change hoti hai, isliye dono edge radii single value

y

par collapse ho jaati hain; tiny difference higher-order small hota hai aur limit mein drop out ho jaata hai.

W4. Radius factor "axis se doori" kyun hai, sirf ek coordinate kyun nahi?

Ek point ek circle sweep karta hai jiska circumference

2 π

times uski spinning axis se doori hai; coordinate sirf tab us doori ke barabar hoti hai jab axis corresponding coordinate axis ho.

W5. In integrals mein substitution itni baar kyun aati hai (jaise

u = 1 + 4 x^{2}

1 + (y^{'})^{2}

mein aksar ek inner function chhupa hota hai jiska derivative already bahar radius factor ke roop mein hota hai, isliye

u

-substitution puri cheez ko clean

\int u d u

mein collapse kar deta hai — Integration by substitution dekho.

W6. Sphere derivation mein constant integrand

r

kyun milta hai?

Height

y = r^{2} - x^{2}

aur arc-slant

\frac{r}{r ^{2} - x ^{2}}

multiply hote hain to awkward roots cancel ho jaate hain, sirf

r

bachta hai — yeh sphere ki uniform curvature ki signature hai.

W7. Parametric curves ke liye

1 + (d y / d x)^{2}

ki jagah

(d x / d t)^{2} + (d y / d t)^{2}

kyun use karte hain?

Jab dono coordinates

t

ke saath move karte hain, arc length,

d x

aur

d y

par jointly Pythagoras hai;

1 + (y^{'})^{2}

form sirf

d x

factor out karne ke baad yahi hai, jo tab fail hota hai jab

d x / d t = 0

ho (vertical tangent). Parametric curves dekho.

Edge cases

E1. Jahan curve rotation axis ko touch karti hai, wahan surface area contribution exactly kya hai?

Zero, kyunki wahan radius

0

hai isliye

2 π r d s \to 0

; ring ek single point par degenerate ho jaati hai aur surface smoothly ek tip par close ho jaati hai.

E2. Agar curve mein vertical tangent ho (

d y / d x \to \infty

) to integrand ka kya hoga?

x

-form mein

1 + (y^{'})^{2}

blow up ho jaata hai, jo signal deta hai ki

y

mein integrate karo (use

1 + (d x / d y)^{2}

) jahan slope finite ho.

E3. Kya surface area finite ho sakti hai agar curve infinity tak stretch ho?

Haan —

[1, \infty)

par ek curve (jaise Gabriel's Horn ki boundary) divergent surface de sakti hai jabki ek related quantity finite rahe;

S

converge karega ya nahi, yeh depend karta hai ki

y d s

kitni tez decay karta hai.

E4. Ek single point ki surface area of revolution kya hai?

Zero. Ek point ki koi arc length nahi hoti (

d s = 0

), isliye sweep karne ke liye kuch nahi — woh zyada se zyada ek circle (ek curve) trace karta hai, jiska area zero hota hai.

E5. Agar ek curve poori tarah rotation axis par lie kare, to woh kaunsi surface sweep karti hai?

Koi area wali surface nahi — har point ki radius

0

hai, isliye poori cheez spin hokar axis par ek line segment mein aa jaati hai,

2 π \cdot 0 \cdot d s = 0

contribute karte hue.

E6. Ek straight vertical segment (constant

x = c > 0

y

from

0

h

) ko y-axis ke baare mein rotate karo — kaunsi shape aur area milti hai?

Radius

c

, height

h

ka ek cylinder; yahan

d s = d y

aur

S = \int_{0}^{h} 2 π c d y = 2 π c h

, jo familiar "unrolled rectangle" lateral area hai.

E7. Usi vertical segment ko x-axis ke baare mein rotate karo — ab kya hoga?

Yeh ek flat annular disk (washer) sweep karta hai, lateral band nahi; surface-of-revolution lateral formula yahan apply nahi hota kyunki piece axis ke parallel hai — ise flat ring ke roop mein treat karo jiska area

π (x_{2}^{2} - x_{1}^{2})

hai.

E8. Jab curve closed aur symmetric ho, jaise ek full circle ko ek diameter ke baare mein rotate kiya jaaye, to

S

kya hoga?

Sphere milta hai; lekin poori boundary integrate karni hogi aur double-counting se bachna hoga — upar aur neeche wale semicircles usi surface ko trace karte hain, isliye unhe poori width par ek baar use karo.

Connections

Surface area of revolution — parent note jise ye traps drill karte hain.
Arc length — woh $d s$ jo "slant not shadow" trap banata hai.
Volume of revolution (disk & shell) — TF3 mein $π r^{2}$ vs $2 π r d s$ ka contrast.
Frustum and cone geometry — average-radius rule ka source (SE4, W2).
Parametric curves — vertical-tangent edge case (E2, W7).
Integration by substitution — $u$ -subs kyun aate hain (W5).
Pappus's theorem — arc-centroid vs region-centroid (SE6).