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

4.2.3 · Maths › Calculus II — Integration

WHY hume yeh chahiye?

Ek rectangle ka area trivial hai: width × height. Ek curve ka area nahi. Integration ki poori trick hai ek mushkil area ko infinitely many aasaan areas se badalna. "Riemann sum" finite approximation hai; "definite integral" uski limit hai.

WHAT hoti hai ek partition? (scaffolding banao)

WHAT hota hai ek Riemann sum? (general form)

Ek hi freedom hai — height kahan se sample karo. Teen classic choices:

Rule	Sample point $x_{i}^{*}$	Height aati hai…
Left	$x_{i - 1} = a + (i - 1) Δ x$	strip ke left edge se
Right	$x_{i} = a + i Δ x$	strip ke right edge se
Midpoint	$\frac{x _{i - 1} + x _{i}}{2} = a + (i - \frac{1}{2}) Δ x$	strip ke centre se

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

DERIVATION: scratch se formal definite integral

WHY limit lete hain? Har $S_{n}$ sirf ek approximation hai. Jab hum zyada, patli rectangles use karte hain ( $n \to \infty$ , $Δ x \to 0$ ) toh yeh staircase curve ko aur tightly hug karta hai. Exact area limiting value hai.

Step 1 — "mesh" define karo. Partition ka mesh (norm) sabse wide strip hai: $∥ P ∥ = max_{i} Δ x_{i}$ . Yeh step kyun? $∥ P ∥ \to 0$ karna har strip ko vanish hone par majboor karta hai, sirf average ko nahi.

Step 2 — limit ka choices se independent hona demand karo.

"Har choice" kyun? Agar left, right, aur midpoint sums sab usi number par converge karte hain, toh answer humari arbitrary sampling par depend nahi kar sakta — yeh $f$ ki ek genuine property hai, hamare method ki nahi.

Step 3 — uniform-partition shortcut. Continuous $f$ ke liye ek uniform partition kaafi hai, aur $∥ P ∥ \to 0 ⟺ n \to \infty$ : $\int_{a}^{b} f (x) d x = n \to \infty lim i = 1 \sum n f (a + i Δ x) \frac{b - a}{n}, Δ x = \frac{b - a}{n} .$

Notation ab khud explain karta hai: $\int$ ek stretched "S" hai Sum ke liye; $d x$ , $Δ x$ ki limit hai (ek infinitely thin width); $f (x)$ height hai.

Worked Example 1 — $\int_{0}^{1} x^{2} d x$ definition se

Hum ise right sums ki limit ke roop mein compute karte hain.

$Δ x = \frac{1 - 0}{n} = \frac{1}{n}$ , aur $x_{i} = 0 + i \cdot \frac{1}{n} = \frac{i}{n}$ . Yeh step kyun? Right rule right edge $x_{i}$ ko sample karta hai.

$R_{n} = \sum_{i = 1}^{n} (\frac{i}{n})^{2} \cdot \frac{1}{n} = \frac{1}{n ^{3}} \sum_{i = 1}^{n} i^{2} .$ Yeh step kyun? Constants $\frac{1}{n ^{2}} \cdot \frac{1}{n} = \frac{1}{n ^{3}}$ bahar nikaalo, pure $\sum i^{2}$ bachta hai.

$\sum_{i = 1}^{n} i^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}$ use karo: $R_{n} = \frac{1}{n ^{3}} \cdot \frac{n ( n + 1 ) ( 2 n + 1 )}{6} = \frac{( n + 1 ) ( 2 n + 1 )}{6 n ^{2}} .$ Yeh step kyun? Closed form plug karo taaki hum clean limit le sakein.

$\int_{0}^{1} x^{2} d x = lim_{n \to \infty} \frac{2 n ^{2} + 3 n + 1}{6 n ^{2}} = \frac{2}{6} = \frac{1}{3} .$ Yeh step kyun? Top aur bottom ko $n^{2}$ se divide karo; $\frac{3}{n}, \frac{1}{n ^{2}}$ terms mar jaate hain. ✓ ( $\frac{x ^{3}}{3}_{0}^{1}$ se match karta hai).

Worked Example 2 — numerical L, R, M for $\int_{1}^{3} \frac{1}{x} d x$ , $n = 2$

True value $= ln 3 - ln 1 = ln 3 \approx 1.0986$ . $Δ x = \frac{3 - 1}{2} = 1$ .

Left ( $x_{0} = 1, x_{1} = 2$ ): $L_{2} = (f (1) + f (2)) \cdot 1 = (1 + 0.5) = 1.5$ . Kyun? Left edges $1$ aur $2$ sample karo.

Right ( $x_{1} = 2, x_{2} = 3$ ): $R_{2} = (f (2) + f (3)) \cdot 1 = (0.5 + 0. \overline{3}) = 0.8 \overline{3}$ . Kyun? Right edges $2$ aur $3$ sample karo.

Midpoint ( $1.5, 2.5$ ): $M_{2} = (f (1.5) + f (2.5)) \cdot 1 = (0. \overline{6} + 0.4) = 1.0 \overline{6}$ . Kyun? Har strip ke centres sample karo.

Notice karo ki $\frac{1}{x}$ decreasing hai, toh $L >$ truth $> R$ , aur $M \approx 1.067$ sabse close hai $1.0986$ ke. Exactly wahi jo predict kiya tha. ✓

Worked Example 3 — symbolic midpoint for $\int_{0}^{2} (3 x + 1) d x$

Linear function ⇒ midpoint kisi bhi $n$ ke liye exact hai. $n = 1$ lo: $Δ x = 2$ , midpoint $= 1$ . $M_{1} = f (1) \cdot 2 = (3 \cdot 1 + 1) \cdot 2 = 8.$ True: $\frac{3 x ^{2}}{2} + x_{0}^{2} = 6 + 2 = 8$ . ✓ Exact kyun? Ek line par midpoint height ke upar cut off hone wala triangle neeche add hone wale triangle ke equal hota hai — perfect cancellation.

Common Mistakes (Steel-manned)

Recall Feynman: ek 12-saal ke bacche ko explain karo

Socho tum ek hill-shaped khet ka area nikalna chahte ho lekin tumhare paas sirf ek ruler hai. Tum khet ko patli vertical strips mein kaatate ho aur pretend karte ho ki har strip ek flat rectangle hai. Rectangle ki height lene ke liye tum strip mein ek jagah choose karte ho — uski left side, right side, ya middle — aur wahan hill ki height measure karte ho. Har height ko strip ki width se multiply karo, sab rectangles ko add karo. Jitna patla kaatoge, tumhara answer real area ke utna close hoga. "Definite integral" sirf woh perfect answer hai jo tum tab paate ho jab slices infinitely thin ho jaati hain.

Active Recall

[a, b]

ki partition kya hoti hai?

Ek set

a = x_{0} < x_{1} < \dots < x_{n} = b

jo interval ko subintervals

[x_{i - 1}, x_{i}]

mein width

Δ x_{i}

ke saath todta hai.

Riemann sum ka general formula?

S_{n} = \sum_{i = 1}^{n} f (x_{i}^{*}) Δ x_{i}

, jahan

x_{i}^{*}

[x_{i - 1}, x_{i}]

mein koi bhi sample point hai.

LEFT rule ka sample point (uniform)?

x_{i - 1} = a + (i - 1) Δ x

RIGHT rule ka sample point?

x_{i} = a + i Δ x

MIDPOINT rule ka sample point?

a + (i - \frac{1}{2}) Δ x

Increasing

f

ke liye, left sum over- ya under-estimate hai?

Underestimate (right overestimate hai).

\int_{a}^{b} f d x

ki formal definition?

Common limit

lim_{∥ P ∥ \to 0} \sum f (x_{i}^{*}) Δ x_{i}

jo har sample points ki choice ke liye same ho, jab yeh exist kare.

Mesh

∥ P ∥

kya hai?

Sabse bada subinterval width,

max_{i} Δ x_{i}

Uniform-partition integral formula?

lim_{n \to \infty} \sum_{i = 1}^{n} f (a + i Δ x) \frac{b - a}{n}

jahan

Δ x = \frac{b - a}{n}

Limit sample-independent kyun honi chahiye?

Taaki value

f

ki property ho, haari arbitrary rectangle heights ki nahi.

\int_{0}^{1} x^{2} d x

right sums se compute karo.

lim_{n} \frac{1}{n ^{3}} \frac{n ( n + 1 ) ( 2 n + 1 )}{6} = \frac{1}{3}

Konsa rule linear functions ke liye exact hai?

Midpoint rule (over/under triangles cancel ho jaate hain).

Integral mein

d x

kya represent karta hai?

Strip width

Δ x \to 0

ki limit.

Ek aisi function do jo Riemann integrable nahi hai.

Dirichlet function

1_{Q}

— sample-dependent limits.

\int x^{2}

ke liye use ki gayi sum identity?

\sum_{i = 1}^{n} i^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}

Connections

Fundamental Theorem of Calculus — is limit ko antiderivative evaluation $F (b) - F (a)$ mein convert karta hai.
Trapezoidal Rule — left aur right sums ka average.
Simpson's Rule — zyada accuracy ke liye weighted blend (midpoint + trapezoid).
Summation formulas — $\sum i, \sum i^{2}, \sum i^{3}$ limit evaluation ke liye zaroori hain.
Area under a curve aur Signed area — integral kya measure karta hai.
Continuity and Integrability — limit exist hone ke liye sufficient conditions.
Limits of sequences — $n \to \infty$ ki machinery.