4.2.15
Ek pyaaz ko ulte tarike se chhilne ki kalpana karo: hum ek solid ko patli cylindrical "shells" (nested tin cans ki tarah) se banate hain. Har shell ek patli tube hai. Agar hum saari tubes ke volumes ko jod dein, toh hume solid of revolution ka total volume milta hai.
Key trick yeh hai: ek patli shell ko agar cheera jaaye aur납chapta kiya jaaye, toh woh sirf ek rectangular sheet ban jaati hai. Uska volume = (wall ki lambai) × (height) × (thickness).
Intuition Disks hamesha kyun nahi use karte?Disk/washer method solid ko rotation ke axis ke perpendicular slice karta hai. Yeh tab bahut achha kaam karta hai jab har slice ek clean disk ho. Lekin kabhi kabhi y x
Shell method axis ke parallel slice karta hai. Isse tum uss variable mein integrate karte reh sakte ho jo tumhari curve ke liye already natural hai. WHY yeh powerful hai: tum aksar function ko invert karne se bachte ho.
Intuition Can ko unroll karoEk patli shell lo jiska radius r h d x
iski width = shell ki circumference = 2 π r
iski height = h
iski thickness = d x
Worked example Example 1 — basic,y y = x 2 y = 0 x = 2 y
Step 1 — strip orientation chuno. Vertical strips ⇒ shells y Kyun: axis y Step 2 — radius & height. r = x h = f ( x ) = x 2 Kyun: y x y = 0 y = x 2 Step 3 — integral set up karo. x 0 2 V = ∫ 0 2 2 π x ⋅ x 2 d x = 2 π ∫ 0 2 x 3 d x . Step 4 — integrate karo. = 2 π [ 4 x 4 ] 0 2 = 2 π ⋅ 4 = 8 π . x = y
Worked example Example 2 — vertical linex = 3 y = x 2 y = 0 x = 2 x = 3
Step 1 — radius. r = 3 − x Kyun: axis x = 3 daayein taraf hai (0 ≤ x ≤ 2 = 3 − x Step 2 — height. abhi bhi h = x 2 Step 3 — integral.
V = ∫ 0 2 2 π ( 3 − x ) x 2 d x = 2 π ∫ 0 2 ( 3 x 2 − x 3 ) d x . Step 4 — integrate karo. = 2 π [ x 3 − 4 x 4 ] 0 2 = 2 π ( 8 − 4 ) = 8 π .
Worked example Example 3 — do curves ke beechRegion y = x y = x 2 x = 0 , 1 y
Step 1 — kaun upar hai? ( 0 , 1 ) x > x 2 h = x − x 2 Yeh step kyun: height strip ka top minus bottom hota hai.
Step 2 — radius. r = x Step 3 — integral.
V = ∫ 0 1 2 π x ( x − x 2 ) d x = 2 π ∫ 0 1 ( x 2 − x 3 ) d x . Step 4. = 2 π [ 3 x 3 − 4 x 4 ] 0 1 = 2 π ( 3 1 − 4 1 ) = 2 π ⋅ 12 1 = 6 π .
Common mistake "Radius hameshax Yeh sahi kyun lagta hai: textbook ke pehle example mein r = x Fix: r doori hai. x = c ∣ x − c ∣ x
Common mistake Horizontal axis ke baare mein rotate karte waqtf ( x ) Yeh sahi kyun lagta hai: f ( x ) Fix: jab axis horizontal ho, toh shells horizontal hote hain, thickness d y horizontal width x right − x left parallel chalti hain.
2 π Yeh sahi kyun lagta hai: disks π r 2 π Fix: shell ki circumference 2 π r 2 π
d x y Fix: agar thickness d x sab kuch (radius, height, limits) x
Recall Integrate karne se pehle predict karoy = x 2 0 ≤ x ≤ 2 x = 3 Forecast: kya volume y x = 0 dono = 8 π mila — hairan kar dene waala! Badhaa hua radius aur geometry milke kaam karte hain. Compute karke verify karo — guess par kabhi andha bharosa mat karo. (Yeh "steel-man your mistake" wali muscle hai.)
Shell method volume element d V = 2 π r h d x
Shells mein 2 π tube ko unroll karne par flat sheet milti hai jiska width = circumference = 2 π r
Axis x = c r = ∣ x − c ∣
Strip orientation rule strips rotation ke axis ke parallel chalti hain
y f ( x ) V = ∫ a b 2 π x f ( x ) d x
Shells ko disks se kab prefer karte hain jab inverse function solve karna mushkil ho; natural variable rakhte hain
Do curves ke beech ke region ki height h = ( top ) − ( bottom )
Derivation: d x 2 π h ( 2 r d x + d x 2 ) d x 2
Horizontal line y = c d y r = ∣ y − c ∣ h = x R − x L
y = x 2 , y = 0 , x = 2 y 8 π
Recall Feynman: ek 12-saal ke bachche ko explain karoTin cans ka ek stack socho, sab alag alag sizes ke, ek doosre ke andar baithe hue Russian dolls ki tarah — sabse bada bahar, sabse chhota andar. Har can bahut patla hai. Agar tum sirf ek can lo, use side se kaato, aur납flat karo, toh tumhe ek flat rectangle milta hai. Iski area hai "kitna lamba hai chaaron taraf" times "kitna uuncha," aur yeh patla hai, toh iski volume yeh area times patlaapan hai. Ab un saare납납납flatten kiye hue cans ke volumes ko jodo aur tumne poore pyaaz jaisi shape ka solid naap liya. Yeh jodna integral hai, aur "chaaron taraf kitna lamba" 2 π r r 2 π r
Mnemonic Formula yaad karo"2-Pi Radius, Height, Thickness" → 2 π r h d x "Round the Tube, Up the Wall, Across the Skin" ke roop mein bolo.
Round (circumference 2 π r h d x
Volume of revolution — disk and washer method — perpendicular-slice wala cousin; jo bhi f Definite integral as a limit of Riemann sums — "infinitely many shells ko add karo" kyun kaam karta hai.Area between two curves — "top − bottom" height wala idea reuse hota hai.Arc length and surfaces of revolution — agla step: ek curve ko revolve karo, solid nahi surface milti hai.Choosing dx vs dy strips — orientation ka decision generalize hota hai.
slit and unrolled becomes
avoids inverting function
radius r, height h, thickness dx
Exact limit: gap between two cylinders
V = integral 2 pi x f of x dx
Disk-washer method slices perpendicular
Shell slices parallel to axis
Vertical line x = c uses radius abs x minus c
Horizontal line y = c uses thickness dy
