Visual walkthrough — Area between curves — horizontal and vertical slices

4.2.13 · D2 · Maths › Calculus II — Integration › Area between curves — horizontal and vertical slices

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Step 1 — "Area between curves" ka matlab kya hai

KYA HAI. Hamare paas do curves hain. Upar wali ko hum $y=f(x)$ bulayenge; neeche wali ko $y=g(x)$. Yahan $y=f(x)$ sirf ek shorthand hai "jab tum position $x$ par khade ho toh upar wali curve ki height kitni hai" — $x$ matlab kitna daayein ho, $y$ matlab kitna upar ho. Shaded region woh sab kuch hai jo unke beech mein ghira hua hai.

YEH YAHAN SE KYUN SHURU KAREIN. Kisi bhi integral se pehle, humein pakka karna hai ki hum kaunsi quantity dhoondh rahe hain: shaded pocket ka area, na kisi ek curve ke neeche ka area. Top aur bottom ko ab naam dena hume baad mein sign ki galtiyon se bachata hai.

PICTURE. Do curves aur unke beech ka shaded pocket. Dhyan do: kisi bhi horizontal position $x$ par, pocket ki ek top edge hai (curve $f$ par) aur ek bottom edge (curve $g$ par).

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Step 2 — Pocket ko khade rectangles mein kaat lo

KYA HAI. Pocket ke andar koi ek position $x$ chuno aur wahan ek bahut patli vertical strip kaat lo. Use ek tiny width do. Us tiny width ko hum $\Delta x$ kehte hain — Greek letter $\Delta$ ("delta") "mein ek chhoti si change" ka standard signal hai, toh $\Delta x$ matlab "horizontal position mein ek chhoti si change." Yeh strip (lagbhag bilkul sahi) ek seedha khada rectangle hai.

SLICE KYUN KAREIN. Tedhe-meedhe kinaron wale pocket ka koi area formula nahi hota. Lekin rectangle ka hota hai: length times width. Toh hum ek naasambhav shape ko hazaron aasaan rectangles ki dheri se badal dete hain — yahi to Definite Integral as Riemann Sum ke peeche ka poora idea hai.

PICTURE. Pocket ke andar khada ek laal rectangle, jiska width $\Delta x$ labelled hai, uska top $f$ ko chhoo raha hai aur bottom $g$ ko.

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Step 3 — EK rectangle ko napo (height aur width)

KYA HAI. Laal rectangle ki:

• height = top edge minus bottom edge $=f(x)-g(x)$,
• width = $\Delta x$.

Toh uska area hai

$$\underset{\text{height}=\text{top }y-\text{bottom }y}{\underbrace{(f(x)-g(x))}}\cdot \underset{\text{tiny width}}{\underbrace{\Delta x}}.$$

SUBTRACT KYUN KAREIN? Height upar ki taraf naapi jaati hai. Strip ka top height $f(x)$ par baitha hai; bottom height $g(x)$ par. Do heights ke beech ki doori badi minus chhoti hoti hai, bilkul waise jaise kisi kursi ke upar tumhari height (tumhare sar ki height) $-$ (kursi ki height) hoti hai. Kyunki humne declare kiya tha ki $f\ge g$, yeh gap $f(x)-g(x)$ kabhi negative nahi hoga — theek hai, kyunki length negative nahi ho sakti.

PICTURE. Ek rectangle ka zoom: $g(x)$ se upar $f(x)$ tak ek vertical measuring arrow labelled "$f(x)-g(x)$", aur $\Delta x$ labelled ek horizontal brace.

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Step 4 — Saare rectangles ko jodo (Riemann sum)

KYA HAI. Poori pocket ko $n$ aisi strips se, ek doosre ke saath, $x=a$ se $x=b$ tak bharo. Unhe $i=1,2,\dots ,n$ number karo. Strip $i$ kisi sample position $x_i^{\ast }$ par baithti hai (chhota sa star bas matlab "woh particular $x$ jo maine strip $i$ ke andar chuna"). Saralata ke liye har strip ko same width $\Delta x=\frac{b-a}{n}$ do (poore span $b-a$ ko $n$ barabar tukdon mein kaatna). Har strip ka area jodne par:

$$A\approx \sum _{i=1}^n\underset{\text{height of strip }i}{\underbrace{(f(x_i^{\ast })-g(x_i^{\ast }))}}\underset{\text{its width}}{\underbrace{\Delta x}}.$$

Symbol ${\sum }_{i=1}^n$ (capital Greek "sigma") matlab hai "neeche diye hue ko jodo, counter $i$ ko $1$ se $n$ tak chalate hue." Yeh sirf "strip $1$ + strip $2$ + $\dots$ + strip $n$" likhne ka compact tarika hai.

APPROXIMATELY EQUAL KYUN? Har strip ka top thoda tilted hota hai lekin humne use flat treat kiya. Toh hum thode se off hain — tops par tiny triangular gaps ya overhangs reh jaate hain.

PICTURE. Kaafi saare laal rectangles se bhari pocket; top edge ke saath chhoti si error slivers dikhaai de rahi hain aur daayein taraf sikit'ti ja rahi hain jahan strips patli khinchi hain.

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Step 5 — Width ko zero karo: integral paida hota hai

KYA HAI. Ab strips ko patla aur zyada karo: $n\to \infty$ karo, jo force karta hai $\Delta x\to 0$. Error slivers gayab ho jaati hain aur approximation exact ban jaati hai. Is sum ki limit hai hi — by definition — integral:

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

Right-hand side ko term by term padho:

• $\int$ = "continuous jodna" (ek stretched $S$ for Sum),
• $a$ (neeche) aur $b$ (upar) = jahan sweep shuru aur khatam hoti hai,
• $f(x)-g(x)$ = position $x$ par rectangle ki height (top minus bottom),
• $dx$ = ab infinitely patli width (woh cheez jo $\Delta x$ ban gayi).

YEH TOOL KYUN AUR SIMPLE MULTIPLICATION NAHI? Height $f(x)-g(x)$ badlti rehti hai jab $x$ move karta hai — tum "height × width" ek baar nahi kar sakte. Integral precisely woh machine hai jo ek varying height ko width se multiply karke total karne ke liye banayi gayi hai. Is limit ko ek actual number mein banana Fundamental Theorem of Calculus ka kaam hai.

PICTURE. Do panels: mote strips (top ke saath dikhaai dene wale steps) ek taraf, ultra-fine strips (top bilkul smooth dikhta hai) doosri taraf — $\Delta x\to 0$ ka visual.

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Step 6 — $a$ aur $b$ kahan se aate hain? (pinch points)

KYA HAI. Limits $a$ aur $b$ woh $x$-values hain jahan do curves milti hain. Unhe dhundne ke liye, donon heights ko equal karo, $f(x)=g(x)$, aur solve karo — aksar ek quadratic ya polynomial hoti hai, dekho Solving Quadratic & Polynomial Equations.

KYUN. Meeting point par top aur bottom ek saath aa jaate hain, toh pocket ki height zero hoti hai wahan — woh pinch ho ke band ho jaati hai. Yahi region ki natural left aur right edge hai.

PICTURE. Curves do dots par cross karti hain; region har dot par kuch nahi reh jaata, beech mein mota hota hai.

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Step 7 — Khatarnaak case: curves jo andar cross karti hain

KYA HAI. Maan lo curves beech mein role swap karti hain: $f$ kuch der top par rehta hai, phir crossing point $x=c$ ke baad $g$ upar aa jaata hai. Agar tum blindly ${\int }_a^b(f-g)dx$ integrate karo, toh woh hissa jahan $g$ ab upar hai ek negative amount contribute karta hai, jo asli area ka kuch hissa cancel kar deta hai.

YEH KYUN FAIL KARTA HAI. $f-g$ left piece par positive hai aur right piece par negative; integral unhe signs ke saath jodta hai, toh woh ek doosre ko kha jaate hain. Area, magar, hamesha positive hota hai — cancellation allowed nahi.

FIX. Crossing $c$ par split karo aur hamesha har piece par (top $-$ bottom) likho:

$$A={\int }_a^c(f-g)dx+{\int }_c^b(g-f)dx={\int }_a^b|f-g|dx.$$

$|\cdots |$ ("absolute value") matlab hai "ise positive kar do," jo sirf ek shorthand hai "top minus bottom, jo bhi abhi top par hai."

PICTURE. Curves $c$ par cross karti hain; left region shaded jahan $f$ upar hai, right region shaded jahan $g$ upar hai — do pockets, har ek apne sahi top-minus-bottom ke saath naapi gayi.

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Step 8 — Apna sir ghoomao: horizontal slices ($dy$)

KYA HAI. Kuch regions khade strips ke liye awkward hote hain kyunki top curve khud badal jaati hai jab tum daayein sweep karte ho (jaise ek sideways parabola $x=y^2$, jo ek $x$ ke liye do $y$'s deti hai). Ilaaj: rectangles ko lita do. Ab ek strip ki ek fixed height $y$ hai, ek tiny thickness $\Delta y$ ("vertical position mein ek chhoti si change"), curve $x=R(y)$ par ek right edge aur $x=L(y)$ par ek left edge. Yahan $R(y)$ matlab "jab tum height $y$ par khade ho toh right curve kitni daayein hai." Steps 2–5 ko $90^{\circ }$ ghuma ke repeat karo:

$$A={\int }_c^d(==R(y)-L(y)==)dy.$$

• $R(y)-L(y)$ = ek lete hue rectangle ki length (right $x$ minus left $x$),
• $dy$ = uski tiny thickness,
• $c,d$ = woh $y$-values jahan curves milti hain (ab $y$ mein solve karo).

KYUN. Geometry kabhi nahi badli — bas woh direction badla jisme hum sweep karte hain. $y=f(x)$ ko $x=(\text{something in }y)$ mein rewrite karne ke liye tum Inverse Functions use karte ho. Yeh wohi slice-measure-add pattern hai jo hum Volumes by Slicing & Disks mein ek dimension upar reuse karenge.

PICTURE. Wohi pocket, ab horizontal laal rectangles se bhari; ek highlighted hai left-se-right measuring arrow "$R(y)-L(y)$" aur thickness $\Delta y$ ke saath.

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Ek-picture summary

Sab kuch ek single mantra aur ek single diagram mein aa jaata hai: slicing direction chuno, ek rectangle napo, aur integral ko infinitely many jodne do.

Summary figure kaise padho. Figure mein do panels hain jo usi pocket ko dikhate hain:

• Left panel (vertical slices). Grey shape Steps 1–6 ki pocket hai. Khada laal rectangle Step 3 ki tile hai: uski height top $-$ bottom $=f(x)-g(x)$ hai aur width $\Delta x$ hai. Aise tiles ko left→right sweep karke jodna hi integral ${\int }_a^b(f-g)dx$ hai jo uske upar print hua hai.
• Right panel (horizontal slices). Wohi pocket, ab ek lete hue laal rectangle se bhari jo Step 8 se hai: uski length right $-$ left $=R(y)-L(y)$ hai aur thickness $\Delta y$ hai. Bottom→top sweep karna deta hai ${\int }_c^d(R-L)dy$.
• Woh punchline jo do panels visible karti hain: wahi grey region, do sweep directions, ek area — slicing ki direction kabhi bhi answer nahi badal sakti.

Recall Feynman: poora walkthrough ek 12-saal ke bachche ko batao

Do lahraati lines ka picture socho jinke beech ek coloured pocket hai. Tum pocket ko directly naap nahi sakte, toh use bahut patli sticks ke bunch mein kaat lo. Agar sticks khadi hain, toh har ek ki height hai "top line us jagah bottom line se kitni upar hai" aur width tiny hai — us width ko $dx$ bulao. Ek stick ke liye height times width karo, phir har stick ko jodo: yeh jodna hi woh hai jo $\int$ symbol karta hai. Pocket ke kinar (jahan se shuru karke jodna band karo) bas woh spots hain jahan do lines milti hain aur pocket ko pinch karke band karti hain. Agar lines beech mein cross karti hain, toh ek stick ka "top" "bottom" ban jaata hai, toh kaam ko crossing par split karo aur hamesha talle wali minus chhoti wali lo. Aur agar pocket ko leti hui sticks se bharna aasaan ho, toh wahi karo — length napo "right line minus left line" thickness $dy$ ke saath. Wohi pocket, wohi area, bas sticks sidewise.

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Connections

• Definite Integral as Riemann Sum — Steps 4–5 is idea ka action mein whi hain.
• Fundamental Theorem of Calculus — integral set up hone ke baad use evaluate karta hai.
• Inverse Functions — horizontal slices ke liye $y=f(x)$ ko $x=R(y)$ mein rewrite karta hai (Step 8).
• Solving Quadratic & Polynomial Equations — pinch points dhundta hai (Step 6).
• Volumes by Slicing & Disks — wohi slice-measure-add pattern, ek dimension upar.