Surface area of revolution — Hinglish notes

4.2.17 · Maths › Calculus II — Integration

Hum kya compute kar rahe hain?

KYA add karte hain: infinitely many patli bands ka lateral (side) area. KYUN band aur disk nahi: hum surface chahte hain, toh har slice apna outer edge contribute karta hai, circumference $2 π r$ ka ek ring.

Formula ko scratch se derive karna

Step 1 — Ek frustum ka area (truncated cone)

Slant height $L$ aur base radius $r$ wale full cone ka lateral area $π r L$ hota hai (use unroll karke sector banao). Frustum do cones ka difference hota hai; algebra se milta hai:

$A_{frustum} = π (r_{1} + r_{2}) ℓ$

jahan $ℓ$ band ki slant height hai. Yeh step kyun? Kyunki $\frac{r _{1} + r _{2}}{2}$ average radius hai — toh yeh "average circumference $\times$ slant length" hai, bilkul unrolled cylinder ki tarah.

Step 2 — Band ko infinitesimal banao

Ek tiny piece ke liye, $r_{1} \approx r_{2} \approx y$ (curve ki height), aur slant length $ℓ$ arc-length element $d s$ ban jaati hai. Toh:

$d S = 2 π y d s$

$2 π y$ kyun? Yeh us circle ki circumference hai jo point $y$ trace karta hai. $d s$ kyun, $d x$ nahi? Kyunki surface tilted curve ke along wrap hota hai, uske flat shadow ke along nahi.

Step 3 — $d s$ (arc length element) ko express karo

Ek tiny right triangle par Pythagoras se, jiske legs $d x$ aur $d y$ hain:

$d s = d x^{2} + d y^{2} = 1 + (\frac{d y}{d x})^{2} d x$

Step 4 — Integrate karo

Worked Examples

Common Mistakes (Steel-man + Fix)

Recall Feynman: 12-saal ke bachche ko explain karo

Socho tum ek spinning top ki poori surface cover karne ke liye uske around ek ribbon wrap kar rahe ho. Top ki outline ko tiny slanted stairs mein kaat lo. Har tiny stair, jab spin hoti hai, ek patla ring banata hai. Ek ring ka area hai "kitna door around" ( $2 π \times$ middle pole se distance) times "slanted edge kitna lamba hai." Saare rings add karo aur tumne poori shape paint kar di. Woh trick jo sab bhool jaate hain: slanted edge length use karo, flat floor length nahi — kyunki surface jhukti hai!

Active Recall

X-axis ke baare mein y=f(x) ke liye surface area formula

S = \int_{a}^{b} 2 π y 1 + (y^{'})^{2} d x

Surface area mein

d x

ki jagah

d s

kyun use karte hain

Kyunki surface tilted curve ke along wrap hota hai;

d s

true arc-length element hai,

d x

sirf uska horizontal shadow hai

Radii

r_{1}, r_{2}

slant

ℓ

wale frustum ka lateral area

π (r_{1} + r_{2}) ℓ

(average circumference × slant)

Y-axis ke baare mein rotate karne par radius factor

x

(y-axis se distance), giving

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

Arc length element

d s

in terms of dx

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

Revolution se radius r ke sphere ka surface area

4 π r^{2}

Cone radius r slant L ka lateral surface area

π r L

X-axis ke baare mein parametric surface area

S = \int 2 π y (x^{'})^{2} + (y^{'})^{2} d t

Line y=k ke baare mein rotate karne par radius

∣ y - k ∣

Connections

Arc length — yahan $d s$ bilkul arc-length integrand jaisa hi hai.
Volume of revolution (disk & shell) — volume mein $π r^{2}$ use hota hai, surface mein $2 π r d s$ .
Frustum and cone geometry — $π (r_{1} + r_{2}) ℓ$ ka source.
Parametric curves — radius aur $d s$ ko generalize karna.
Integration by substitution — inme se zyaadatar integrals evaluate karne ke liye use hota hai.
Pappus's theorem — $S = 2 π \overset{r}{ˉ} L$ , surface = centroid ki circumference × arc length.