4.2.18 · D1Average value of a function
Infinitely many function values ka average nikalne ke liye, hum unhe sab ek integral se add karte hain (ek "continuous sum") aur interval ki length se divide karte hain. Toh yeh poora topic sirf do ideas se bana hai jo ek saath jude hain: summing (jo integral ban jaata hai) aur dividing by how many (jo width b − a
Yeh page kuch bhi assume nahi karta. Parent note par tumhe ek formula milega jo ek function ko average karta hai; "f f avg us answer ka naam samjho jise hum build kar rahe hain. Isse pehle ki tum parent note par woh formula dekho, hum us mein har ek symbol ek ek karke build karenge, har ek ke saath ek picture aur ek reason ki woh wahan kyon hona zaroori hai.
Ek function ek machine hai: tum ise ek number x f ( x ) f x f ( x )
Ise ek curve ki tarah picture karo. Horizontal axis par har point ek input x height output f ( x )
Intuition Yeh topic ko yeh kyon chahiye"Average value" un saari heights ka average hai. Agar tum f ( x )
Itne-saare-values kyon matter karte hain Ek function ke paas interval mein har x
Definition Closed interval[ a , b ] [ a , b ] a b a left endpoint (start) hai aur b right endpoint (end) hai. Hum hamesha b > a
Picture mein: a b beech ka curve ka hissa hi woh hissa hai jis par hum dhyan dete hain.
Intuition Yeh topic ko yeh kyon chahiyeTum sirf ek definite stretch par hi average kar sakte ho. "x 2 x 2 [ 0 , 3 ]
Definition Interval ki widthb − a length hai — start a b b > a positive hota hai.
Picture: shaded region ke base ki length, horizontal axis ke along measure ki gayi.
Intuition Yeh topic ko yeh kyon chahiyeJab tum n n width b − a
b − a a − b Kyon tempt karta hai: pressure mein letters ka order ulajh jaata hai.
Kyon galat hai: ek width negative nahi ho sakti. a − b b > a Fix: bada minus chhota. Width = b − a > 0
Ab sabse bada symbol. Ise pieces mein build karte hain.
Intuition "Area" "sum" ka answer kyon deta haiSocho ki shaded region ko thin vertical strips mein kaato. Har strip (almost) ek rectangle hai: uski height f ( x ) width ek tiny number hai jise hum d x × = f ( x ) d x Har strip ka area add karo aur tumhare paas total area hoga. Woh "saari strips ka area add karo" wali baat hi stretched-S symbol ∫ Sum ke liye lambe "S" ke roop mein shuru hua tha.
Definition Definite integral, symbol by symbol∫ a b f ( x ) d x
∫ a b f ( x ) height .d x infinitely thin width .Saath mein: curve aur horizontal axis ke beech ka total signed area , a b
"Signed" matter karta hai: axis ke neeche waali strips (jahan f ( x ) negative area count hoti hain. §6 dekho.
Intuition Yeh topic ko yeh kyon chahiyeIntegral "infinitely many heights add karo" wala step hai. Average value = yeh total, evenly spread. Integral ke bina infinitely many heights sum karne ka koi tarika nahi hai — Definite Integral as Area aur Riemann Sums dekho ki sum kaise rigorous banayi jaati hai.
Parent ki derivation isse hoke gujarti hai, toh har symbol earn karte hain.
Definition Sigma notation∑ i = 1 ∑ n f ( x i ) f ( x 1 ) + f ( x 2 ) + ⋯ + f ( x n ) ∑ Sum ke liye); i 1 n x i i
Definition Har sample pointx i [ a , b ] n Δ x = n b − a i ek input chunte ho — use x i
strip ka left endpoint : x i = a + ( i − 1 ) Δ x
right endpoint : x i = a + i Δ x midpoint , ya literally strip ke andar koi bhi point.
f ( x i )
Intuition Kyon choice limit mein matter karna band kar deti haiAlag alag choices thodi alag staircases deti hain (figure dekho: left-endpoint bars vs right-endpoint bars). Lekin jaise strips thinner hoti jaati hain (n → ∞ saari choices same integral par converge karti hain . Yahi reason hai ki notation ∑ f ( x i ) Δ x x i
Δ x n
n Δ x = n b − a Δ
Definition Yeh limit kab exist karna guaranteed hai? (integrability)Staircase sum tabhi ek single number par settle hoga jab function well-behaved ho. Iss topic mein hum jo safe, sufficient condition use karte hain woh hai:
f is == continuous == on [ a , b ] . integrable hota hai, toh uske Riemann sums ka limit hota hai aur ∫ a b f d x
Common mistake Yeh assume karna kihar function ka average value hota hai
Kyon tempt karta hai: formula aisa lagta hai ki hamesha kaam karta hai.
Kyon galat hai: agar f Fix: pehle check karo ki f [ a , b ]
Intuition Yeh topic ko yeh kyon chahiyeYeh derivation mein bridge hai. Finite average n 1 ∑ f ( x i ) b − a 1 ∑ f ( x i ) Δ x ∫
Tumhe har case handle karna hai, sirf un curves ko nahi jo axis ke upar rehti hain.
Definition Height ka sign
f ( x ) > 0 upar hai → area positive count hota hai.f ( x ) < 0 neeche hai → area negative count hota hai.f ( x ) = 0
Intuition Yeh topic ko yeh kyon chahiyeAverage value negative ho sakta hai (agar curve zyaatar axis ke neeche baithti hai) ya exactly zero (agar positive aur negative parts cancel kar lein). Example: f ( x ) = sin x [ 0 , 2 π ] 0
Isse pehle ki hum woh example compute karein, hum ek aur tool chahte hain — koi aisa tarika ki ek integral ko actually evaluate kar sakein bina infinitely many strips haath se sum kiye.
Definition Antiderivative aur evaluation bracketf antiderivative koi bhi function F f F f − cos x sin x − cos x sin x
Fundamental Theorem of Calculus phir kehta hai: total signed area antiderivative ko b a evaluation bracket se likhte hain:
∫ a b f ( x ) d x = [ F ( x ) ] a b = F ( b ) − F ( a ) . [ ⋯ ] a b
Worked example Ek zero-average caseF ( x ) = − cos x sin x ∫ 0 2 π sin x d x = [ − cos x ] 0 2 π = ( − cos 2 π ) − ( − cos 0 ) = ( − 1 ) − ( − 1 ) = 0. f avg = 2 π 1 ⋅ 0 = 0
n → ∞ lim ( expression ) n paas se paas tum jaate ho, bina kabhi infinity par pahunche.
Picture: §5 ke rectangles ki staircase dekho. n = 4 n = 20 n = 1000
Intuition Yeh topic ko yeh kyon chahiye"Infinitely many ka average" kuch aisa nahi hai jo hum directly compute kar sakein — hum n limit lete hain . Limit hi hai jo ek approximation ko exact continuous average mein upgrade karta hai.
Definition "Average" ke liye notationKisi letter ke upar ek bar, jaise y ˉ f ˉ f avg
Intuition Yeh topic ko yeh kyon chahiyeYeh sirf ek label hai taaki hum answer ke baare mein baat kar sakein bina har baar poora formula rewrite kiye.
Ab
f avg = b − a 1 ∫ a b f ( x ) d x f avg b − a 1 ∫ a b f ( x ) d x f Area over width — kuch bhi unexplained nahi bachta.
Foundations is order mein stack hoti hain. Ise padhο as "neeche wala box zaroori hai upar wale box ke kaam karne se pehle":
Function as a height f ( x ) interval [ a , b ] Interval se width b − a
Interval ko kaatna aur sampled heights sum karna Riemann sum ∑ f ( x i ) Δ x sample-point choice aur continuity/integrability ki zaroorat hai well-defined hone ke liye.
Limit n → ∞ definite integral ∫ a b f d x antiderivative bracket (§6) se evaluate hota hai.Width se divide karke integral average value f avg Mean Value Theorem for Integrals power karta hai.
Recall Self-test: kya tum peek karne se pehle har ek ka jawab de sakte ho?
f ( x ) Function f x x height hai.
Interval [ a , b ] a b b > a
Width b − a a − b Width positive hoti hai; kyunki b > a = b − a > 0
∫ a b f ( x ) d x ∫ a b f ( x ) d x ∫ a , b f ( x ) d x
Integral ek "sum" kyun compute karta hai? Har thin strip ka area f ( x ) d x
Har strip ke andar sample point x i Strip mein kahin bhi — left endpoint, right endpoint, midpoint, ya koi bhi point; limit mein choice matter nahi karti.
f f [ a , b ]
∑ i = 1 n f ( x i ) Δ x n → ∞ Definite integral ∫ a b f ( x ) d x
Δ x n Δ x = n b − a n
Evaluation bracket [ F ( x ) ] a b F ( b ) − F ( a ) F f
lim n → ∞ Woh single number jis par expression settle hota hai jaise n
f ( x ) < 0 Yeh negative area count hota hai, jo average ko zero se neeche khींch sakta hai ya positive parts ko zero par cancel kar sakta hai.
[ 0 , 2 π ] sin x 0
