Foundations — Riemann sums — left, right, midpoint; formal definition of definite integral

4.2.3 · D1 · Maths › Calculus II — Integration › Riemann sums — left, right, midpoint; formal definition of d

Pehle aapko the main note ka ek bhi line padhne se pehle har ek symbol apna banana hoga. Ye page har ek ko scratch se build karta hai — seedhi baatein, phir ek picture, phir kyun is topic ko uski zaroorat hai. Har item apne pehle wale par tika hai, isliye upar se neeche padhein.

0. Graph — ek curve actually hota kya hai

Ek horizontal number line (the $x$-axis) aur ek vertical line (the height) imagine karo. Har $x$ par function batata hai ki curve kitna ooncha khada hai. Poora subject is region ke baare mein hai jo is curve aur $x$-axis ke beech mein trap hai — neeche shaded blueprint dekho.

Kyun is topic ko iski zaroorat hai: "area under a curve" ka koi matlab nahi jab tak aap na jaano ki curve sirf "height as a function of position" hai. Woh shaded region woh prize hai jiske peeche hum hain. Dekho bhi Area under a curve.

1. Interval $[a,b]$ — jahaan hum measure karte hain

$x$-axis par do vertical fence-posts imagine karo: ek $x=a$ par, ek $x=b$ par. Hum sirf un posts ke beech ke area ki parwah karte hain.

Kyun is topic ko iski zaroorat hai: ek area ke edges hone chahiye. $[a,b]$ exactly batata hai ki region kahaan shuru aur kahaan khatam hoti hai.

2. Subscripts $x_0,x_1,\dots ,x_n$ — bahut saare points ko label karna

$a$ aur $b$ ke beech khade fence-posts ki ek row imagine karo. Har ek ke liye naye letters invent karne ki jagah, hum $x$ reuse karte hain aur usmein ek numbered tag lagate hain: post number $0$, post number $1$, post number $n$ tak.

Kyun is topic ko iski zaroorat hai: hum $[a,b]$ ke across bahut saare cutting points lagaenge aur har ek ko individually refer karna hoga. Subscripts naming system hain.

3. $n$ aur partition — interval ko slice karna

Figure dekho: $a$ se $b$ tak ki fence ko upright cutting lines se $n$ vertical strips mein kaata gaya hai.

Kyun is topic ko iski zaroorat hai: ek curvy region mushkil hai; bahut saari patli strips har ek almost ek rectangle hain. Partition kaatne ka act hai. Linked idea: Continuity and Integrability.

4. $\Delta x$ — ek strip ki width

Do neighbouring cutting lines ke beech ki horizontal distance imagine karo — woh base length $\Delta x$ hai. Agar poori fence $b-a$ wide hai aur aapne use $n$ equal pieces mein kaata, toh har piece total divided by $n$ hai: woh hai $\frac{b-a}{n}$.

Kyun is topic ko iski zaroorat hai: ek rectangle ka area width × height hai. $\Delta x$ woh width hai. Isse miss karo aur aap areas ki jagah heights add kar rahe hain.

5. Sample point $x_i^{\ast }$ — jahaan hum height read karte hain

Hamare paas teen natural spots hain — strip ka left edge, right edge, ya middle. Figure usi strip ke liye teeno heights dikhata hai.

Kyun is topic ko iski zaroorat hai: ek rectangle ko ek height chahiye, lekin curve ki height ek strip mein vary karti hai. Sample point haara decision hai ki kaunsi single height pe bharosa karein.

6. Summation $\sum$ — ek saath bahut saare terms add karna

Kyun is topic ko iski zaroorat hai: hamare paas $n$ rectangles total karne hain. $f(x_1^{\ast })\Delta x+f(x_2^{\ast })\Delta x+\dots$ likhna unbearable hai; $\sum$ ek baar mein bol deta hai. Handy closed forms Summation formulas mein hain.

Ab poora Riemann sum decode hota hai: $S_n={\sum }_{i=1}^n\underset{\text{height}}{\underbrace{f(x_i^{\ast })}}\underset{\text{width}}{\underbrace{\Delta x_i}}=\text{total area of all }n\text{ rectangles.}$

7. Limit $\underset{n\to \infty }{\lim }$ — exact answer tak squeeze karna

Figure dikhata hai ki $n$ double hone par staircase kitni tight hoti jaati hai.

Kyun is topic ko iski zaroorat hai: integral is limit ke roop mein defined hai — ye woh exact area hai jiske paas rectangles sirf approach kar sakte hain. Background: Limits of sequences.

8. Integral sign ${\int }_a^{b}f(x)dx$ — finished notation

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

Kyun is topic ko iski zaroorat hai: yahi destination hai — har pehla symbol isme funnel hota hai. Ek baar jab aap $\int$ ko "$\sum$ taken to the limit" ke roop mein dekh lete ho, notation magic nahi lagta. Ise evaluate karne ka shortcut Fundamental Theorem of Calculus mein hai; refinements Trapezoidal Rule aur Simpson's Rule hain.

Foundations topic ko kaise feed karte hain

Har box exactly ek symbol hai jo aapne abhi seekha; arrows ka matlab hai "left wale ko samajhne ke liye right wale ki zaroorat hai".

Equipment checklist

Right side cover karo aur khud ko test karo — aap main note ke liye tabhi ready hain jab har line automatic ho.