Arc length formula — derivation: Question bank — Hinglish deep dive

4.2.16 · D5 · Maths › Calculus II — Integration › Arc length formula — derivation

Woh poora formula jise hum stress-test kar rahe hain: $L = \int_{a}^{b} 1 + (\frac{d y}{d x})^{2} d x, d s = d x^{2} + d y^{2} .$ Poore time chota right triangle dimag mein rakho: base $d x$ , height $d y$ , hypotenuse $d s$ .

True or false — justify karo

Arc length un do endpoints ko join karne wali straight line se chhoti ho sakti hai.

False. Ek straight line do points ke beech ka shortest path hoti hai, isliye curve — har chord ki apni straight piece se kam se kam utni hi lambaai — hamesha

\geq

endpoint chord hoti hai (dekho Pythagorean theorem).

Agar ek curve

x = a

x = b

tak horizontal hai (

y

constant), toh uski arc length exactly

b - a

hogi.

True. Tab

d y / d x = 0

, toh integrand

1 + 0 = 1

hai aur

L = \int_{a}^{b} 1 d x = b - a

— ek flat curve bas apna base hi hoti hai.

Ek curve ki har

y

-value ko double karne se uski arc length bhi double ho jaati hai.

False. Vertically stretch karne se slope

d y / d x

square root ke andar nonlinearly badalta hai, isliye length bhi badlti hai lekin 2 ke clean factor se nahi.

Curve ko upar shift karna (

y \to y + c

) uski arc length nahi badalta.

True. Vertical shift

d y / d x

nahi badlta (constant ka derivative 0 hota hai), isliye integrand aur length bilkul same rehte hain.

Arc length integrand

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

hamesha

\geq 1

hota hai.

True.

(y^{'})^{2} \geq 0

, toh root ke andar wali cheez kam se kam 1 hai, matlab arc hamesha horizontal distance se kam se kam utni hi tezi se accumulate hoti hai.

x = g (y)

wali curve ke liye hamesha

\int_{a}^{b} 1 + (d y / d x)^{2} d x

use kar sakte hain.

False. Agar

x = g (y)

mein vertical tangents hain (jaise sideways parabola), toh

d y / d x

blow up karta hai; tumhe

y

mein integrate karna hoga:

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

Arc length hamesha tab badhti hai jab interval

[a, b]

badhaate hain.

True. Integrand

1 + (y^{'})^{2} \geq 1 > 0

hai, isliye bade interval par integrate karne se strictly positive amount ki length add hoti hai.

Mean Value Theorem wala step require karta hai ki

f

[a, b]

par har jagah differentiable ho.

Zyaadatar true aur respect ke laayak: Mean Value Theorem ko

f

ki closed interval par continuity aur open interior par differentiability chahiye; ek corner (non-differentiable point) clean single formula ko tod deta hai aur integral ko split karne par majboor karta hai.

Error dhundho

L = \int_{a}^{b} y d x

, kyunki hum totals paane ke liye integrate karte hain."

\int y d x

curve ke neeche ka area calculate karta hai, distance along it nahi. Length mein hypotenuse se aaya

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

hona zaroori hai;

y

akela height measure karta hai, travel nahi.

L = \int_{a}^{b} ∣ f^{'} (x) ∣ d x

— slope batata hai ki tum kitna door jaate ho."

Yeh sirf vertical travel

Δ y

count karta hai aur horizontal leg

Δ x

ko ignore karta hai. Root ke andar wala "

1

" exactly wahi missing

d x

term hai; ise drop karo aur har slanted step chhoti ho jaati hai.

(\frac{x}{4} + \frac{1}{x})^{2} = \frac{x}{4} + \frac{1}{x}

for all

x

Sirf tab valid hai jab andar wali cheez

\geq 0

ho. Generally

a^{2} = ∣ a ∣

; absolute value drop karne se pehle actual interval par sign check karna zaroori hai.

(Δ x)^{2} + (Δ y)^{2} = Δ x + Δ y

Square root ko sum ke upar split nahi kar sakte;

a^{2} + b^{2} \neq = a + b

. Woh galat step Pythagoras ko ignore karta aur hamesha hypotenuse overestimate karta.

"Humne

Δ x_{i}

(Δ x_{i})^{2} = Δ x_{i}

se bahar nikala — koi absolute value nahi chahiye, toh yeh lucky simplification hai."

Yeh luck nahi hai: humne labels is tarah choose kiye ki

x_{i - 1} < x_{i}

, jisse

Δ x_{i} > 0

banta hai, isliye

(Δ x_{i})^{2} = Δ x_{i}

genuinely sach hai. Positivity by construction hai, accident se nahi.

"Riemann sum

\sum 1 + (f^{'} (x_{i}^{*}))^{2} Δ x_{i}

mein

x_{i}^{*}

left endpoint par choose karna zaroori hai."

Nahi — Mean Value Theorem humein ek specific interior point

x_{i}^{*}

deta hai, aur kyunki limit mein koi bhi tag point kaam karta hai jab

f^{'}

continuous ho, sum usi integral par converge karta hai.

"Sharp corner wali curve ki arc length wahan infinite hoti hai."

False. Corner

f^{'}

ko ek point par undefined banata hai lekin length finite rehti hai — bas integral ko corner par split karo aur do finite pieces add kar lo.

Why questions

Square root ke andar "

+ 1

" kyun aata hai, "

+ 2

" ya kuch nahi kyun?

Kyunki

d s^{2} = d x^{2} + d y^{2}

;

d x^{2}

se divide karne par

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

milta hai, aur woh akela

1

exactly horizontal-leg contribution

d x^{2} / d x^{2}

hai.

Hum straight chords se approximate kyun karte hain, maan lo tiny circular arcs se kyun nahi?

Ek straight segment hi aisi single shape hai jiska length hum pehle se exactly jaante hain (Pythagoras). Hum woh tool use karte hain jo hamare paas hai, phir tiles ko shrink karte hain taaki approximation error khatam ho jaaye.

Mean Value Theorem ek integration derivation mein kyun aata hai?

Yeh measurable secant slope

Δ y /Δ x

ko actual derivative value

f^{'} (x_{i}^{*})

mein convert karta hai, jisse sum differences ki mess ki jagah

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

ka Riemann sum ban jaata hai.

f^{'}

continuous kyun hona chahiye taaki limit ek definite integral bane?

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

ki Riemann integrability chahiye ki integrand accha (nice) ho (continuous

f^{'}

kaafi hai); ek wildly discontinuous slope limiting sum ko integral par converge karna fail kar sakta hai.

Wahi

d s

, surfaces of revolution mein reuse kyun hota hai?

Kyunki surface of revolution arc elements ko spin karke banti hai: har strip ki length

d s

hoti hai aur circumference

2 π (radius)

, isliye

d A = 2 π r d s

wahi element reuse karta hai jo humne yahan derive kiya.

Wahi curve

y = f (x)

x = g (y)

dono tarah kyun likhi ja sakti hai aur same length de sakti hai?

Length path ki ek geometric property hai, independent of which variable you sweep; do integrals sirf identical

d s

ke alag-alag parametrizations hain, jaisa ki parametric form clearly dikhata hai.

Formula vertical tangent par kyun fail hota hai (ya care kyun chahiye)?

Wahan

d y / d x \to \infty

, isliye

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

x

mein integrand ke roop mein diverge karta hai;

y

mein integrate karne par switch karo, jahan slope

d x / d y

finite hota hai.

Edge cases

Zero-width interval

a = b

par arc length kya hogi?

Zero.

\int_{a}^{a} (\dots) d x = 0

— koi interval nahi, koi path nahi, koi length nahi.

Integrand

f^{'} (x) = 0

wale point par (flat spot / local max) kya hota hai?

Wahan

1 + 0 = 1

hota hai; curve momentarily horizontal hai, isliye woh pure horizontal rate par length accumulate karta hai — perfectly finite aur well behaved.

Ek curve ka

[a, b]

mein kahin vertical tangent (slope

\to \infty

) hai. Kya hum phir bhi uski length find kar sakte hain?

Haan, lekin wahan

x

mein integrate karke nahi.

x = g (y)

ke roop mein reparametrize karo ya parameter $t$ use karo, taaki infinite slope ek finite, integrable expression ban jaaye.

Kya arc length formula polar coordinates mein di gayi curve jaise

r = 1 + cos θ

par kaam karta hai?

Directly nahi — tumhe polar version chahiye

L = \int r^{2} + (d r / d θ)^{2} d θ

, jo usi

d s^{2} = d x^{2} + d y^{2}

se convert karke derive hota hai.

Agar

f^{'}

x = c \in (a, b)

par jump discontinuity (corner) hai, toh

L

kaise compute karte hain?

Split karo:

L = \int_{a}^{c} + \int_{c}^{b}

. Har piece smooth aur finite hai; unhe add karo. Akela corner khud koi length contribute nahi karta.

Agar curve wapas dobaara jaati hai (

x

ki function nahi), jaise ek poora circle?

y = f (x)

formula vertical-line test fail karta hai. Parametric arc length use karo, jahan

x

aur

y

dono kisi parameter

t

par depend karte hain, taaki path freely loop kar sake.

Recall Har trap ka one-line summary

Length hypotenuse $d s = d x^{2} + d y^{2}$ mein rehti hai: kabhi $y$ nahi (woh area hai), kabhi akela $∣ y^{'} ∣$ nahi (woh sirf vertical hai), kabhi $∣ \cdot ∣$ drop mat karo (sign check), aur kabhi $x$ mein nahi jahan tangent vertical ho jaaye (variables switch karo).

Connections

Arc length formula — derivation — woh parent jise yeh bank interrogate karta hai.
Pythagorean theorem — $d s$ ki woh shape kyun hai.
Mean Value Theorem — secant-to-derivative bridge jo upar test ki gayi.
Riemann sums and the definite integral — limiting sum integral kyun hai.
Parametric arc length, Polar arc length, Surface area of revolution — jahan edge cases aage le jaate hain.