4.4.20
Intuition Ek line mein ideaEk triple integral ∭ E f d V f E d V add karta hai . Poora game yeh hai: (1) region ko boxes mein kaato, (2) aisi coordinates choose karo jo boxes aur region ko describe karna aasaan bana de. ==Cartesian= seedhi divaaren=, ==cylindrical= ek plane mein gol + seedha axis=, ==spherical= har direction mein gol=.
Definition Triple integral (Riemann definition)Ek solid E N Δ V k ( x k ∗ , y k ∗ , z k ∗ ) ∭ E f ( x , y , z ) d V = lim N → ∞ ∑ k = 1 N f ( x k ∗ , y k ∗ , z k ∗ ) Δ V k .
YEH kya compute karta hai:
Agar f = 1 ∭ E d V = Volume ( E )
Agar f = ρ ( x , y , z ) ∭ E ρ d V = Mass
General f
WHY yeh kuch useful mein converge karta hai: har term f ⋅ Δ V k
HOW limits set karte hain (yahi ek skill jo matter karti hai):
Innermost (z x , y E z = z 1 ( x , y ) enter karta hai aur z = z 2 ( x , y ) exit karta hai. Yeh (x , y Middle (y E x y y y 1 ( x ) , y 2 ( x ) Outer (x constant range [ x 1 , x 2 ]
Common mistake "Mere outer limits inner variable ke functions kyun hain?"Yeh natural lagta hai ki ∫ 0 x … d x outer limits constants hone chahiye , aur har limit sirf unhi variables pe depend kar sakta hai jo uske bahar hain. Reason: tum andar se bahar integrate karte ho; jab outer integral tak pahuncho, baaki saare variables ja chuke hote hain, isliye sirf ek number reh sakta hai.
Worked example Tetrahedronx , y , z ≥ 0 , x + y + z ≤ 1 Step 1 — inner z Arrow upar: z = 0 z = 1 − x − y Kyun? Tilha hua face x + y + z = 1 ∫ 0 1 − x − y d z = 1 − x − y Step 2 — middle y x y x , y ≥ 0 , x + y ≤ 1 y 0 → 1 − x Kyun? Constraint mein z = 0 ∫ 0 1 − x ( 1 − x − y ) d y = [ ( 1 − x ) y − 2 y 2 ] 0 1 − x = 2 ( 1 − x ) 2 Step 3 — outer x x : 0 → 1 ∫ 0 1 2 ( 1 − x ) 2 d x = 2 1 ⋅ 3 1 = 6 1 6 1
Intuition Boxes warp ho jaate hain, isliyed V ( x , y , z ) ( u , v , w ) d u d v d w warped parallelepiped mein map ho jaata hai. Uska volume = (box ka volume) × (local stretch). Woh stretch ∣ J ∣ Jacobian determinant .
x = r cos θ , y = r sin θ , z = z , r ≥ 0 , 0 ≤ θ < 2 π .
Tab use karo jab: region ya integrand mein ek axis ke around rotational symmetry ho (cylinders, cones, paraboloids).
r ≤ a 0 ≤ z ≤ h
=\int_0^{2\pi}\!\!d\theta\int_0^a r\,dr\int_0^h dz
=2\pi\cdot\tfrac{a^2}{2}\cdot h=\pi a^2 h.\ \checkmark$$
**Product mein kyun split kiya?** Limits saari constants hain aur integrand factor ho jaata hai, isliye Fubini 1D integrals multiply karne deta hai.
x = ρ sin ϕ cos θ , y = ρ sin ϕ sin θ , z = ρ cos ϕ , ρ ≥ 0 0 ≤ ϕ ≤ π + z 0 ≤ θ < 2 π z
sin ϕ sin θ do angular arc-lengths hain. Latitude circles poles pe (ϕ = 0 , π shrink hote hain, aur sin ϕ → 0 ϕ θ
a
=\underbrace{\int_0^{2\pi}d\theta}_{2\pi}\;\underbrace{\int_0^\pi\sin\phi\,d\phi}_{2}\;\underbrace{\int_0^a\rho^2 d\rho}_{a^3/3}
=2\pi\cdot2\cdot\tfrac{a^3}{3}=\tfrac{4}{3}\pi a^3.\ \checkmark$$
**Yeh 80/20 example kyun hai:** yeh teeno factors simultaneously check karta hai aur famous $\frac43\pi a^3$ pe land karta hai.
∭ E z d V ρ ≤ a , z ≥ 0 z = ρ cos ϕ z ≥ 0 ⇒ ϕ ∈ [ 0 , π /2 ] ∫ 0 2 π ∫ 0 π /2 ∫ 0 a ( ρ cos ϕ ) ρ 2 sin ϕ d ρ d ϕ d θ Step kyun: integrand ρ 3 cos ϕ sin ϕ
=2\pi\cdot\tfrac12\cdot\tfrac{a^4}{4}=\tfrac{\pi a^4}{4}.$$
Use kiya $\int_0^{\pi/2}\sin\phi\cos\phi\,d\phi=\tfrac12$.
Region / symmetry
Best coords
d V
Box, planes, slanted faces
Cartesian
d x d y d z
Circle/cylinder/cone/paraboloid kisi axis ke around
Cylindrical
r d r d θ d z
Sphere, ball, x 2 + y 2 + z 2
Spherical
ρ 2 sin ϕ d ρ d ϕ d θ
Common mistake Sphere pe Cartesian use karnaYeh safe lagta hai (koi Jacobian yaad nahi karna). Lekin z = ± a 2 − x 2 − y 2 x 2 + y 2 + z 2 ρ = a
Recall Khud test karo (answers hide hain)
Har coordinate system mein d V
Cylindrical mein r
ρ 2 sin ϕ ρ 2 Outer limits constants kyun hone chahiye?
Recall Feynman: ek 12-saal ke bacche ko explain karoSocho tum ek shape ko tiny sugar cubes se bharte ho aur count karte ho ki har cube mein kitna "stuff" (meethaas) hai. Sab add karo — woh ek triple integral hai. Agar shape ek box hai, square cubes use karo (Cartesian). Agar woh ek tin can hai, usse curved tiles mein slice karo jo bahar ki taraf fan out hote hain — center se door wale tiles wider hote hain, isliye hum multiply karte hain ki woh kitne door hain (r shorter bhi ho jaate hain, isliye hum sin ϕ
d V "Cylinders carry ONE r ρ 2 sin ϕ
Spherical mantra: "rho-squared sine-phi" — do ρ sin ϕ poles ko kill karta hai .
What is d V d x d y d z
What is d V r d r d θ d z
What is d V ρ 2 sin ϕ d ρ d ϕ d θ
Why does the cylindrical Jacobian equal r A polar pixel ka area r d r d θ r d θ d r d z
Why does spherical d V sin ϕ Latitude-circle arc ka radius ρ sin ϕ ρ sin ϕ d θ
In spherical coords, what are the ranges of ρ , ϕ , θ ρ ≥ 0 0 ≤ ϕ ≤ π + z 0 ≤ θ < 2 π
Volume of a ball of radius a ∫ 0 2 π ∫ 0 π ∫ 0 a ρ 2 sin ϕ d ρ d ϕ d θ = 3 4 π a 3
Volume of cylinder radius a h ∫ 0 2 π ∫ 0 a ∫ 0 h r d z d r d θ = π a 2 h
How do you choose the innermost (z x , y z 1 ( x , y ) z 2 ( x , y )
Why must the outermost integration limits be constants? Tum andar se bahar integrate karte ho; outer step tak baaki saare variables ja chuke hote hain, isliye sirf numbers reh sakte hain.
When should you prefer spherical coordinates? Jab region sphere/ball ho ya integrand mein x 2 + y 2 + z 2 = ρ 2
General change-of-variables formula for d V d V = ∣ det ∂ ( x , y , z ) / ∂ ( u , v , w ) ∣ d u d v d w
What does ∭ E 1 d V E
Cartesian-to-spherical: what is z z = ρ cos ϕ
Triple integral iiint f dV
Vertical arrow enter exit
