Example 25 - Find integral (x2 + 1)dx as limit of a sum - Examples

Example 25 - Chapter 7 Class 12 Integrals - Part 2
Example 25 - Chapter 7 Class 12 Integrals - Part 3 Example 25 - Chapter 7 Class 12 Integrals - Part 4 Example 25 - Chapter 7 Class 12 Integrals - Part 5

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


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Question 1 Find ∫_0^2▒(𝑥^2+1) 𝑑𝑥 as the limit of a sum . ∫_0^2▒(𝑥^2+1) 𝑑𝑥 Putting 𝑎 = 0 𝑏 = 2 ℎ = (𝑏 − 𝑎)/𝑛 = (2 − 0)/𝑛 = 2/𝑛 𝑓(𝑥)=𝑥^2+1 We know that ∫1_𝑎^𝑏▒〖𝑥 𝑑𝑥〗 =(𝑏−𝑎) (𝑙𝑖𝑚)┬(𝑛→∞) 1/𝑛 (𝑓(𝑎)+𝑓(𝑎+ℎ)+𝑓(𝑎+2ℎ)…+𝑓(𝑎+(𝑛−1)ℎ)) Hence we can write ∫_0^2▒(𝑥^2+1) 𝑑𝑥 =(2−0) lim┬(n→∞) 1/𝑛 (𝑓(0)+𝑓(0+ℎ)+𝑓(0+2ℎ)+… +𝑓(0+(𝑛−1)ℎ) =2 lim┬(n→∞) 1/𝑛 (𝑓(0)+𝑓(ℎ)+𝑓(2ℎ)……+𝑓((𝑛−1)ℎ) Here, 𝑓(𝑥)=𝑥^2+1 𝑓(0)=0^2+1=0+1=1 𝑓(ℎ)=ℎ^2+1=(2/𝑛)^2+1=4/𝑛^2 +1 𝑓(2ℎ)=(2ℎ)^2+1=〖4ℎ〗^2+1=4(2/𝑛)^2+1=16/𝑛^2 +1 ….. 𝑓(𝑛−1)ℎ=((𝑛−1)ℎ)^2+1=〖(𝑛−1)^2 (2/𝑛)〗^2+1 =(𝑛−1)^2 × 4/𝑛^2 +1 Hence, our equation becomes = 2 lim┬(n→∞) 1/𝑛 (𝑓(0)+𝑓(ℎ)+𝑓(2ℎ)……+𝑓(𝑛−1)ℎ) = 2 lim┬(n→∞) 1/𝑛 (1+(4/𝑛^2 +1)+(16/𝑛^2 +1" " )+ ……+((4(𝑛 − 1)^2)/𝑛^2 +1)) = 2 lim┬(n→∞) 1/𝑛 ((1 + 1 + 1…𝑛 𝑡𝑖𝑚𝑒𝑠)+0+ 4/𝑛^2 +16/𝑛^2 + …(4(𝑛 − 1)^2)/𝑛^2 ) = 2 lim┬(n→∞) 1/𝑛 (𝑛 +0+ 4/𝑛^2 +16/𝑛^2 + ……(4(𝑛 − 1)^2)/𝑛^2 ) = 2 lim┬(n→∞) 1/𝑛 (𝑛+ 4/𝑛^2 (1+4+ ……+(𝑛 − 1)^2 ) ) = 2 lim┬(n→∞) 1/𝑛 (𝑛+ 4/𝑛^2 (1^2+2^2+ ………+(𝑛 − 1)^2 ) ) = 2 lim┬(n→∞) 1/𝑛 (𝑛+ 4/𝑛^2 ((𝑛 − 1) 𝑛(2𝑛 − 1))/6) = 2 lim┬(n→∞) 1/𝑛 (𝑛+ 4/𝑛 ((𝑛 − 1) (2𝑛 − 1))/6) = 2 lim┬(n→∞) 1/𝑛 (𝑛+ 2/3𝑛 (𝑛−1) (2𝑛−1)) = 2 lim┬(n→∞) (𝑛/𝑛 + 2/(3𝑛^2 ) (𝑛−1) (2𝑛−1)) We know that 1^2+2^2+ ……+𝑛^2= (𝑛(𝑛 + 1) (2𝑛 +1))/6 1^2+2^2+ ……+(𝑛−1)^2= ((𝑛 − 1)(𝑛 − 1+ 1) (2(𝑛 −1)+1))/6 = ((𝑛 − 1) 𝑛(2𝑛 − 1))/6 = 2 lim┬(n→∞) (𝑛/𝑛 + 2/3 ((𝑛 − 1))/𝑛 ((2𝑛 − 1))/𝑛) = 2 lim┬(n→∞) (1+ 2/3 (1− 1/𝑛) (2− 1/𝑛)) = 2 (1+ 2/3 (1−0) (2−0)) = 2 (1+ 2/3 ×2) = 2 (1+ 4/3) = 2 × 7/3 = 𝟏𝟒/𝟑 (lim┬(n→∞) 1/𝑛=0" " )

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.