Question 6 - Area as a sum - Chapter 7 Class 12 Integrals
Last updated at April 16, 2024 by Teachoo
Area as a sum
Question 2 Important Deleted for CBSE Board 2024 Exams
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Question 4 Important Deleted for CBSE Board 2024 Exams
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Question 6 Important Deleted for CBSE Board 2024 Exams You are here
Last updated at April 16, 2024 by Teachoo
Question 6 โซ1_0^4โ(๐ฅ+๐2๐ฅ)๐๐ฅ Let I = โซ1_0^4โ(๐ฅ+๐2๐ฅ)๐๐ฅ I = โซ1_0^4โใ๐ฅ . ๐๐ฅใ+โซ1_0^4โใ ๐2๐ฅ . ๐๐ฅใ Solving I1 and I2 separately Solving I1 โซ1_0^4โใ๐ฅ ๐๐ฅใ Putting ๐ =0 ๐ =4 โ=(๐ โ ๐)/๐ =(4 โ 0)/๐ =4/๐ ๐(๐ฅ)=๐ฅ We know that โซ1_๐^๐โใ๐ฅ ๐๐ฅใ =(๐โ๐) (๐๐๐)โฌ(๐โโ) 1/๐ (๐(๐)+๐(๐+โ)+๐(๐+2โ)โฆ+๐(๐+(๐โ1)โ)) Hence we can write โซ1_0^4โใ๐ฅ ๐๐ฅใ =(4โ0) limโฌ(nโโ) 1/๐ (๐(0)+๐(0+โ)+๐(0+2โ)+โฆ +๐(0+(๐โ1)โ) =4 limโฌ(nโโ) 1/๐ (๐(0)+๐(โ)+๐(2โ)+โฆ +๐((๐โ1)โ) Here, ๐(๐ฅ)=๐ฅ ๐(0)=0 ๐(โ)=โ ๐ (2โ)=2โ ๐((๐โ1)โ)=(๐โ1)โ Hence, our equation becomes โด โซ_0^4โ๐ฅ ๐๐ฅ =4 limโฌ(nโโ) 1/๐ (๐(0)+๐(โ)+๐(2โ)+โฆ +๐((๐โ1)โ) = 4 (๐๐๐)โฌ(๐โโ) 1/๐ (0+โ+2โ+ โฆโฆ+(๐โ1)โ) = 4 (๐๐๐)โฌ(๐โโ) 1/๐ ( โ (1+2+ โฆโฆโฆ+(๐โ1))) We know that 1+2+3+ โฆโฆ+๐= (๐ (๐ + 1))/2 1+2+3+ โฆโฆ+๐โ1= ((๐ โ 1) (๐ โ 1 + 1))/2 = (๐ (๐ โ 1) )/2 = 4 (๐๐๐)โฌ(๐โโ) 1/๐ ( (โ . ๐(๐ โ 1))/2) = 4 (๐๐๐)โฌ(๐โโ) ( ๐(๐ โ 1)โ/2๐) = 4 (๐๐๐)โฌ(๐โโ) ( (๐ โ 1)โ/2) = 4 (๐๐๐)โฌ(๐โโ) ( (๐ โ 1)4/(2 . ๐)) = 4 (๐๐๐)โฌ(๐โโ) ( 2(๐/๐ โ 1/๐)) = 4 (๐๐๐)โฌ(๐โโ) ( 2(1โ 1/๐)) = 4( 2(1โ 1/โ)) [๐๐ ๐๐๐ โ=4/๐] = 4( 2(1โ0)) = 4ร2 = ๐ Solving I2 โซ_0^4โ๐^2๐ฅ ๐๐ฅ Putting ๐ = 0 ๐ =4 โ = (๐ โ ๐)/๐ = (4 โ 0)/๐ = 4/๐ ๐(๐ฅ)=๐^2๐ฅ We know that โซ1_๐^๐โใ๐ฅ ๐๐ฅใ =(๐โ๐) (๐๐๐)โฌ(๐โโ) 1/๐ (๐(๐)+๐(๐+โ)+๐(๐+2โ)โฆ+๐(๐+(๐โ1)โ)) Hence we can write โซ_0^4โ๐^2๐ฅ ๐๐ฅ =(4โ0) limโฌ(nโโ) 1/๐ (๐(0)+๐(0+โ)+๐(0+2โ)+โฆ +๐(0+(๐โ1)โ) =4 limโฌ(nโโ) 1/๐ (๐(0)+๐(โ)+๐(2โ)โฆโฆ+๐((๐โ1)โ) Here, ๐(๐ฅ)=๐^2๐ฅ ๐(0)=๐^(2(0))=1 ๐(โ)=๐^2โ ๐(2โ)=๐^(2(2โ))=๐^4โ ๐((๐โ1)โ)=๐^2(๐โ1)โ Hence we can write โซ_0^4โ๐^2๐ฅ ๐๐ฅ =(4โ0) limโฌ(nโโ) 1/๐ (๐(0)+๐(0+โ)+๐(0+2โ)+โฆ +๐(0+(๐โ1)โ) =4 limโฌ(nโโ) 1/๐ (๐(0)+๐(โ)+๐(2โ)โฆโฆ+๐((๐โ1)โ) Here, ๐(๐ฅ)=๐^2๐ฅ ๐(0)=๐^(2(0))=1 ๐(โ)=๐^2โ ๐(2โ)=๐^(2(2โ))=๐^4โ ๐((๐โ1)โ)=๐^2(๐โ1)โ Hence, our equation becomes โด โซ_0^4โ๐^2๐ฅ ๐๐ฅ =4 limโฌ(nโโ) 1/๐ (๐(0)+๐(โ)+๐(2โ)โฆโฆ+๐(๐โ1)โ) = 4 .limโฌ(nโโ) 1/๐ (1+๐^2โ+๐^4โ+ โฆโฆ+๐^(2(๐ โ 1) โ) ) Let S = 1+๐^2โ+๐^4โ+ โฆโฆ+๐^(2(๐ โ 1) โ) It is a G.P. with common ratio (r) r = ๐^2โ/1 = ๐^2โ We know Sum of G.P = a((๐^๐ โ 1)/(๐ โ 1)) Replacing a by 1 and r by ๐^2โ , we get S = 1(((๐^2โ )^๐ โ 1)/(๐^2โ โ 1))= (๐^2๐โ โ 1)/(๐^2โ โ 1) Thus โด โซ_0^4โ๐^๐ฅ ๐๐ฅ = 4 limโฌ(nโโ) 1/๐ (1+๐^2โ+๐^4โ+ โฆโฆ+๐^(2(๐ โ 1) โ) ) Putting the value of S, we get = 4 .limโฌ(nโโ) 1/๐ ((๐^2๐โ โ 1)/(๐^2โ โ 1)) = 4 (๐๐๐)โฌ(๐โโ) 1/๐ ((๐^2๐โ โ 1)/(2โ . (๐^2โ โ 1)/2โ)) = 4 (๐๐๐)โฌ(๐โโ) (๐^2๐โ โ 1)/2๐โ . 1/( (๐^2โ โ 1)/2โ) = 4 (๐๐๐)โฌ(๐โโ) (๐^2๐โ โ 1)/2๐โ . (๐๐๐)โฌ(๐โโ) 1/( (๐^2โ โ 1)/2โ) Solving (๐ฅ๐ข๐ฆ)โฌ(๐งโโ) ( ๐)/(( ๐^๐๐ โ ๐)/๐๐) As nโโ โ 2/โ โโ โ โ โ0 โด limโฌ(nโโ) ( 1)/(( ๐^2โ โ 1)/2โ) = limโฌ(hโ0) ( 1)/(( ๐^2โ โ 1)/2โ) = 1/1 = 1 Thus, our equation becomes โซ1_0^4โใ๐๐ฅ ๐๐ฅใ ="4" (๐๐๐)โฌ(๐โโ) (๐^2๐โ โ 1)/2๐โ . (๐๐๐)โฌ(๐โโ) 1/( (๐^2โ โ 1)/2โ) " " = "4" (๐๐๐)โฌ(๐โโ) (๐^2๐โ โ 1)/2๐โ . 1 = 4 (๐๐๐)โฌ(๐โโ) (๐^(2๐ . 4/๐) โ 1)/(2๐ (4/๐) ) = 4 (๐๐๐)โฌ(๐โโ) (๐^8 โ 1)/8 (๐๐ ๐๐๐ (๐๐๐)โฌ(๐กโ0) (๐^๐ก โ 1)/๐ก =1) (๐๐ ๐๐๐ โ=4/๐) = 4 (๐^8 โ 1)/8 = (๐^๐ โ ๐)/๐ Putting the values of I1 and I2 in I โด I = โซ1_0^4โใ๐ฅ . ๐๐ฅใ+โซ1_0^4โใ ๐2๐ฅ . ๐๐ฅใ = 8 + (๐^8 โ 1)/2 = (16 + ๐^8 โ 1)/2 = (๐๐ + ๐^๐)/๐