Foundations — Area between curves — horizontal and vertical slices

4.2.13 · D1 · Maths › Calculus II — Integration › Area between curves — horizontal and vertical slices

Is page pe assume kiya gaya hai ki tumne kuch nahi dekha. Hum har wo symbol build karte hain jis pe parent note depend karta hai, ek aisi order mein jahan har ek sirf pehle wale symbols use karta hai. Line one se follow karo.

0. Coordinate plane — jahan sab kuch rehta hai

Kisi bhi curve se pehle, humein ek jagah chahiye. Is topic ki har cheez ek flat sheet pe draw ki jaati hai jisme do number-lines ek point pe cross karti hain jise origin kehte hain.

Topic ko yeh kyun chahiye: har "curve" sirf ek rule hai jo, har us jagah ke liye jo tum choose karo, ek height batata hai. Plane ke bina koi "upar," "neeche," "left," "right" nahi — aur yeh poora chapter is baare mein hai ki hum kis direction mein sweep karte hain.

1. Ek function $y=f(x)$ — ek machine jo $x$ ko height mein badal deti hai

Topic ko yeh kyun chahiye: parent note baar baar kehta hai "top curve $f(x)$, bottom curve $g(x)$." Woh bas do aisi machines hain. Letters $f$ aur $g$ sirf naam hain — aur kuch nahi.

2. Do curves aur unke beech ka gap

Ab do machines ek hi plane pe rakho: $y=f(x)$ (isko top kaho) aur $y=g(x)$ (bottom), jahan region mein har $x$ pe top zyada upar ho, likha jaata hai $f(x)\ge g(x)$.

Topic ko yeh kyun chahiye: $f(x)-g(x)$ poore formula ka star hai. Yeh har vertical slice ki height hai.

3. Interval ko chop karna: $[a,b]$, $\Delta x$, $x_i^{\ast }$

Hum region ko do $x$-values ke beech slice karte hain, $a$ (left edge) aur $b$ (right edge). $a$ se $b$ tak $x$ ka stretch interval $[a,b]$ hai.

Topic ko yeh kyun chahiye: ek patla rectangle ka area hai (height) × (width) $=(f(x_i^{\ast })-g(x_i^{\ast }))\Delta x$. Har rectangle Section 2 ka gap times is thickness se bana hai.

4. Rectangles ko add karna: sum symbol $\sum$

Topic ko yeh kyun chahiye: saare patle rectangles ka total area hai ${\sum }_{i=1}^n(f(x_i^{\ast })-g(x_i^{\ast }))\Delta x.$ Yeh exactly Section 2 ka gap $\times$ Section 3 ki thickness hai, har slice pe add kiya gaya. Yeh idea fully develop kiya gaya hai Definite Integral as Riemann Sum mein.

5. Limit $n\to \infty$ aur integral ${\int }_a^b$

Rectangle-sum sirf approximate hai — uska stair-step top curved boundary ko perfectly hug nahi karta. Isko fix karo infinitely finely chop karke.

Topic ko yeh kyun chahiye: parent note mein har area formula yahi ek limit hai, bas alag top/bottom (ya right/left) curves plug in karke.

6. Integral padhna: Fundamental Theorem shortcut

Tum actually infinitely many cheezein haath se add nahi karte. Iske bajaay:

Topic ko yeh kyun chahiye: yeh woh machine hai jo Section 5 ki scary limit ko arithmetic mein badal deti hai. Poori kahaani Fundamental Theorem of Calculus mein.

7. Viewpoint switch karna: $x=R(y)$, $x=L(y)$, aur inverses

Kabhi kabhi left-to-right ki bajaay bottom-to-top sweep karna zyada clean hota hai. Tab hum rectangles sideways rakhte hain: width ban jaata hai $dy$ (thin height) aur length ban jaata hai right curve − left curve.

Topic ko yeh kyun chahiye: $x=y^2$ jaisi curves ek $x$ ke liye do $y$-values deti hain, toh vertical slicing ko split karna padta hai. Horizontal slicing $x=y^2$ directly padhta hai — koi split nahi.

8. Limits kahan se aate hain? Intersections solve karna

Edges $a,b$ (ya $c,d$) exactly wahan hain jahan do curves milti hain — wahan region pinch ho jaata hai.

Topic ko yeh kyun chahiye: limits ke bina tum integral shuru hi nahi kar sakte, aur limits hain hi intersection points.

Prerequisite map

Equipment checklist

Right side cover karo aur khud ko test karo. Agar koi bhi answer shaky lage, upar woh section dobara padho.

Connections

• Parent topic (Hinglish) — woh note jisme yeh foundations feed hoti hain.
• Definite Integral as Riemann Sum — Sections 3–5 poori detail mein.
• Fundamental Theorem of Calculus — Section 6, antiderivative shortcut.
• Inverse Functions — Section 7, $y=f(x)$ ko $x=L(y)$ ya $x=R(y)$ ke roop mein rewrite karna.
• Solving Quadratic & Polynomial Equations — Section 8, limits find karna.
• Volumes by Slicing & Disks — wahi slice-measure-add pattern, ek dimension upar.