Foundations — Arc length formula — derivation

4.2.16 · D1 · Maths › Calculus II — Integration › Arc length formula — derivation

Yeh page assume karta hai ki aap kuch nahi jaante. Parent note mein derivation ko touch karne se pehle, hum har symbol ko build karte hain — ek ek karke, har ek ek picture se anchored, har ek use hone se pehle earn kiya hua.

0. "Curve $y=f(x)$" ka matlab kya hai

Picture. Ek horizontal axis draw karo (use $x$ kaho, "kitna across") aur ek vertical axis (use $y$ kaho, "kitna upar"). Har $x$ ke liye jahan aap slide karo, rule ek dot mark karta hai height $y=f(x)$ par. Saare dots ko join karo aur aapko ek curve milta hai — ek wiggly line jo $x$-axis positions ke upar rehti hai.

• $a$ aur $b$ $x$-axis par start aur end positions hain. Hum sirf $x=a$ aur $x=b$ ke beech curve ka piece measure karte hain.

1. Do axes aur ek point $(x,y)$

Parent points aise likhta hai $(x_i,y_i)$ aur $(x_{i-1},y_{i-1})$. Unhe aise padho:

• $x_i$ ::: $i$-ve chosen dot ka $x$-address.
• Subscript $i$ sirf ek label counter hai — "dot number $i$." Ismein kuch mysterious nahi hai: $x_0$ pehla hai, $x_1$ doosra, aur aise hi $x_n$ tak, jo aakhri hai.

2. Gap symbols $\Delta x$ aur $\Delta y$

$\Delta x_i=x_i-x_{i-1},\Delta y_i=y_i-y_{i-1}$

Picture. Do neighbouring dots ke beech, ek chota horizontal step aur phir ek chota vertical step draw karo. Horizontal step $\Delta x_i$ hai; vertical step $\Delta y_i$ hai. Dots ko join karne wali straight line ke saath milkar, yeh ek tiny right triangle banate hain.

3. Pythagoras — slanted stick ki length

Ise apne tiny triangle par apply karo. Legs hain $\Delta x_i$ (across) aur $\Delta y_i$ (up); hypotenuse woh straight chord $L_i$ hai jo do dots ko join karta hai:

$L_i=\sqrt{(\Delta x_i{)}^2+(\Delta y_i{)}^2}$

4. Slope: ratio $\frac{\Delta y}{\Delta x}$

Picture. Wahi tiny triangle. Ek near-flat chord mein ek bade across-step par ek chota up-step hota hai → chota ratio. Ek near-vertical chord mein ek bada up-step hota hai → bada ratio.

• Yeh chord-slope, do separated dots use karke, secant slope kehlata hai.

5. Derivative $\frac{dy}{dx}$ aur $f^{\prime }(x)$

Picture. Tiny triangle ko ek point ki taraf shrink karo. Chord curve ke bilkul paas aa jaata hai aur tangent line ban jaata hai — woh straight line jo curve ko wahan sirf kiss karta hai. Iska slope derivative hai.

• Chota prime mark ${}^{\prime }$ $f^{\prime }$ mein sirf matlab hai "$f$ ka derivative." Star $x_i^{\ast }$ mein matlab hai "interval $i$ ke andar koi special point" jiska existence theorem guarantee karta hai.

6. Summation sign $\sum$

${\sum }_{i=1}^n{L}_i=L_1+L_2+\cdots +L_n$

7. Limit $\lim$ aur integral $\int$

${\lim }_{n\to \infty }{\sum }_{i=1}^n\sqrt{1+(f^{\prime }(x_i^{\ast }){)}^2}\Delta x_i={\int }_a^b\sqrt{1+(f^{\prime }(x){)}^2}dx$

• Ek integral ke end mein $dx$ mark karta hai "har infinitely-thin slice ki width," $\Delta x_i$ ka leftover shrinking ke baad.

Prerequisite map

Yeh tools aage kahan jaate hain

Jab $ds=\sqrt{d{x}^2+d{y}^2}$ ka idea yahan solid ho jaata hai, toh wahi element of arc length teen neighbouring topics ko power karta hai: Surface area of revolution ($ds$ ko ek axis ke around spin karo), Parametric arc length (dono $x$ aur $y$ ek clock follow karte hain), aur Polar arc length (angle aur radius se measure karo instead). $ds$ ek baar build karo; har jagah reuse karo.

Equipment checklist

Khud ko test karo — reveal karne se pehle har answer zor se bolo.