4.2.18
Tum jaante ho ki finite numbers ka average kaise nikalte hain: sab jodo, aur kitne hain us se divide karo. Lekin ek function jaise f ( x ) = x 2 a b infinitely many values hoti hain. Infinitely many cheezein ka average kaise nikalte ho?
Trick yeh hai: ek integral ek "continuous sum" hota hai. Toh hum saari function values ko sum karte hain (woh hai ∫ a b f ( x ) d x b − a same area ko hold karta hai.
Definition Finite numbers ka Averagey 1 , y 2 , … , y n y ˉ = n y 1 + y 2 + ⋯ + y n = n 1 ∑ i = 1 n y i
Hum yahi idea [ a , b ] derive karte hain.
Step 1 — Function ko n
[ a , b ] n Δ x = n b − a . Yeh step kyun? Hum sirf finitely many numbers ka average kar sakte hain, isliye hum n f ( x 1 ) , f ( x 2 ) , … , f ( x n )
Step 2 — Un n
f ˉ n = n 1 ∑ i = 1 n f ( x i ) Yeh step kyun? Yeh toh bas ordinary average hai — woh cheez jo hum pehle se jaante hain.
Step 3 — Δ x
Step 1 se, Δ x = n b − a n 1 = b − a Δ x f ˉ n = b − a Δ x ∑ i = 1 n f ( x i ) = b − a 1 ∑ i = 1 n f ( x i ) Δ x Yeh step kyun? Woh sum ∑ f ( x i ) Δ x Riemann sum hai — yeh integral banne ke liye tarsa raha hai.
Step 4 — Limit n → ∞ (zyada se zyada samples = sach mein continuous average):
f avg = n → ∞ lim f ˉ n = b − a 1 ∫ a b f ( x ) d x Yeh step kyun? Jaise n → ∞ Δ x → 0 ban jaata hai, integral ki definition ke anusaar.
Intuition Equal area ka Rectanglef avg [ a , b ] upar nikalte hain exactly woh gaps fill karte hain jo neeche hain.
Definition MVT for IntegralsAgar f continuous hai [ a , b ] c ∈ [ a , b ] f ( c ) = f avg = b − a 1 ∫ a b f ( x ) d x .
c [ a , b ] m M m ≤ f avg ≤ M Intermediate Value Theorem ke anusaar, ek continuous function m M har value leta hai — including f avg f ( c ) = f avg
Worked example Example 1 —f ( x ) = x 2 [ 0 , 3 ] f avg = 3 − 0 1 ∫ 0 3 x 2 d x Kyun? a = 0 , b = 3 = 3 1 [ 3 x 3 ] 0 3 = 3 1 ⋅ 3 27 = 3 1 ⋅ 9 = 3 Yeh step kyun? Power rule se ∫ x 2 d x = x 3 /3 Answer: f avg = 3 Check: f ( c ) = c 2 = 3 ⇒ c = 3 ≈ 1.73 ∈ [ 0 , 3 ]
Worked example Example 2 —f ( x ) = sin x [ 0 , π ] f avg = π − 0 1 ∫ 0 π sin x d x = π 1 [ − cos x ] 0 π Kyun? ∫ sin x d x = − cos x = π 1 ( − cos π + cos 0 ) = π 1 ( 1 + 1 ) = π 2 ≈ 0.637 Note: peak toh 1 ≈ 0.64
Worked example Example 3 — Ek velocity applicationEk car ki velocity v ( t ) = 3 t 2 t ∈ [ 0 , 4 ] v avg = 4 − 0 1 ∫ 0 4 3 t 2 d t = 4 1 [ t 3 ] 0 4 = 4 64 = 16 m/s Yeh kyun important hai: average velocity = (total displacement)/(total time). Aur ∫ 0 4 v d t = 64 hi hai. ✓
Worked example Forecast-then-VerifyPehle forecast karo: f ( x ) = x [ 0 , 10 ] 5 Verify karo: 10 1 ∫ 0 10 x d x = 10 1 ⋅ 2 100 = 5 midpoint par ki value hoti hai — hamesha.
Common mistake "Bas endpoints ka average nikalo:2 f ( a ) + f ( b ) Kyun sahi lagta hai: ek straight line ke liye yeh actually kaam karta hai, aur yahi case hum imagine karte hain.
Kyun galat hai: curved functions ke liye interior values matter karti hain. sin x [ 0 , π ] 2 0 + 0 = 0 π 2 ≈ 0.64 Fix: hamesha integrate karo — endpoints sirf linear f
( b − a ) Kyun sahi lagta hai: integral dekhne mein "answer" lagta hai.
Kyun galat hai: ∫ a b f d x area hai (units: f × x Fix: average = area ÷ width. Hamesha.
b − a 1 a − b Kyun sahi lagta hai: pressure mein sign juggling.
Fix: width positive hoti hai, b > a b − a
Recall Dekhne se pehle try karo: formula batao aur yeh kahan se aata haif avg = b − a 1 ∫ a b f ( x ) d x n 1 ∑ f ( x i ) b − a 1 ∑ f ( x i ) Δ x n → ∞
Recall Feynman: ek 12-saal ke bacche ko samjhaoSocho ek wavy bathtub mein bumpy water level. "Average level" woh hai jahan paani settle ho jaata agar saare bumps flat ho jaate — high wale hisse low wale hisse ko fill karte hain. Ise find karne ke liye, tum total amount of water (area/integral) measure karte ho aur use tub ki length (b − a
Mnemonic Formula yaad karo"Area Over Width." Average = Width of interval Area under curve "Integral ko span se squish karo." (b − a 1 ∫ a b
What is the formula for the average value of f [ a , b ] f avg = b − a 1 ∫ a b f ( x ) d x
Average value kis property wale rectangle ki height hai? Jo rectangle curve ke neeche wale region ke same area ka ho, base [ a , b ]
Hum integral ko ( b − a ) Integral area deta hai; width se divide karna "total" ko "per-unit-length," yaani average height mein convert karta hai.
Yeh formula kahan se derive hota hai? Ordinary average n 1 ∑ f ( x i ) n → ∞
Mean Value Theorem for Integrals batao. Agar f [ a , b ] c ∈ [ a , b ] f ( c ) = b − a 1 ∫ a b f d x
MVT for integrals c f avg min f max f f
Average value of sin x [ 0 , π ] π 2 ≈ 0.637
Average value of f ( x ) = x 2 [ 0 , 3 ] 3
Ek straight line ke liye, average value f Midpoint 2 a + b
Common units check: ∫ a b f d x f x
substitute 1/n = dx over b-a
f_avg = integral over b-a
squeezed between min and max
Area = avg height x width
Intermediate Value Theorem
