Question 5 - Area as a sum - Chapter 7 Class 12 Integrals
Last updated at April 16, 2024 by Teachoo
Area as a sum
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Last updated at April 16, 2024 by Teachoo
Question 5 โซ1_(โ1)^1โใ๐๐ฅ ๐๐ฅใ โซ1_(โ1)^1โใ๐๐ฅ ๐๐ฅใ Putting ๐ =โ1 ๐ =1 โ=(๐ โ ๐)/๐ =(1 โ (โ1))/๐ =(1 + 1)/๐=2/๐ ๐(๐ฅ)=๐^๐ฅ 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_(โ1)^1โใ๐๐ฅ ๐๐ฅใ =(1 โ(โ1)) (๐๐๐)โฌ(๐โโ) 1/๐ (๐(โ1)+๐(โ1+โ)+๐(โ1+2โ)+ โฆ+๐(โ1+(๐โ1)โ)) =2 (๐๐๐)โฌ(๐โโ) 1/๐ (๐(โ1)+๐(โ1+โ)+๐(โ1+2โ)+ โฆ+๐(โ1+(๐โ1)โ)) Here, ๐(๐ฅ)=๐^๐ฅ ๐(โ1)=๐^(โ1) ๐(โ1+โ)=๐^(โ1 + โ) =๐^(โ1). ๐^โ ๐ (โ1+2โ)=๐^(โ1 + 2โ)=๐^(โ1). ๐^2โ ๐(โ1+(๐โ1)โ)=๐^(โ1 + (๐ โ 1)โ)=๐^(โ1).๐^(๐ โ 1)โ Hence, our equation becomes โซ1_(โ1)^1โใ๐๐ฅ ๐๐ฅใ =2 (๐๐๐)โฌ(๐โโ) 1/๐ (๐(โ1)+๐(โ1+โ)+๐(โ1+2โ)+ โฆ+๐(โ1+(๐โ1)โ)) =2 (๐๐๐)โฌ(๐โโ) 1/๐ (๐^(โ1)+๐^(โ1). ๐^โ+๐^(โ1). ๐^2โ+ โฆ+๐^(โ1).๐^(๐โ1)โ ) =2 (๐๐๐)โฌ(๐โโ) 1/๐ (๐^(โ1) (1+โ+๐^2โ+ โฆ+๐^(๐โ1)โ )) =2๐^(โ1) (๐๐๐)โฌ(๐โโ) 1/๐ (1+โ+๐^2โ+ โฆ+๐^(๐โ1)โ ) Let S = 1+โ+๐^2โ+ โฆ+๐^(๐โ1)โ It is G.P with common ratio (r) ๐ = ๐^โ/1 = ๐^โ We know Sum of G.P = a((๐^๐ โ 1)/(๐ โ 1)) Replacing a by 1 and r by ๐^๐ , we get S = 1(((๐^โ )^๐ โ 1)/(๐^๐ โ 1))= (๐^๐โ โ 1)/(๐^๐ โ 1) Thus, โซ1_(โ1)^1โใ๐๐ฅ ๐๐ฅใ =2 . ๐^(โ1) (๐๐๐)โฌ(๐โโ) 1/๐ (1+๐^๐+๐^2โ+ โฆ+๐^(๐โ1)โ ) = 2/๐ (๐๐๐)โฌ(๐โโ) 1/๐ ((๐^๐โ โ 1)/(๐^๐ โ 1)) = 2/๐ (๐๐๐)โฌ(๐โโ) 1/๐ ((๐^๐โ โ 1)/(โ . (๐^โ โ 1)/โ)) = 2/๐ (๐๐๐)โฌ(๐โโ) (๐^๐โ โ 1)/๐โ . 1/( (๐^โ โ 1)/โ) = 2/๐ (๐๐๐)โฌ(๐โโ) (๐^๐โ โ 1)/๐โ . (๐๐๐)โฌ(๐โโ) 1/( (๐^โ โ 1)/โ) Solving (๐ฅ๐ข๐ฆ)โฌ(๐งโโ) ( ๐)/(( ๐^๐ โ ๐)/๐) As nโโ โ 2/โ โโ โ โ โ0 โด limโฌ(nโโ) ( 1)/(( ๐^โ โ 1)/โ) = limโฌ(hโ0) ( 1)/(( ๐^โ โ 1)/โ) = 1/1 = 1 Thus, our equation becomes โซ1_(โ1)^1โใ๐๐ฅ ๐๐ฅใ = 2/๐ (๐๐๐)โฌ(๐โโ) (๐^๐โ โ 1)/๐โ . (๐๐๐)โฌ(๐โโ) 1/( (๐^โ โ 1)/โ) = 2/๐ (๐๐๐)โฌ(๐โโ) (๐^๐โ โ 1)/๐โ . 1 = 2/๐ (๐๐๐)โฌ(๐โโ) (๐^(๐ . 2/๐) โ 1)/(๐ (2/๐) ) = 2/๐ (๐๐๐)โฌ(๐โโ) (๐^2 โ 1)/2 (๐๐ ๐๐๐ (๐๐๐)โฌ(๐กโ0) (๐^๐ก โ 1)/๐ก =1) (๐๐ ๐๐๐ โ=2/๐) = 2/๐ . (๐^2 โ 1)/2 = (๐^2 โ 1)/๐ = ๐^2/๐ โ 1/๐ =๐ โ ๐/๐