Integration Full Chapter Explained - Integration Class 12 - Everything you need

Slide5.JPG

Slide6.JPG
Slide7.JPG Slide8.JPG Slide9.JPG

  1. Chapter 7 Class 12 Integrals
  2. Serial order wise

Transcript

Example 25 Find ∫_0^2β–’(π‘₯^2+1) 𝑑π‘₯ as the limit of a sum . ∫_0^2β–’(π‘₯^2+1) 𝑑π‘₯ Putting π‘Ž = 0 𝑏 = 2 β„Ž = (𝑏 βˆ’ π‘Ž)/𝑛 = (2 βˆ’ 0)/𝑛 = 2/𝑛 𝑓(π‘₯)=π‘₯^2+1 We know that ∫1_π‘Ž^𝑏▒〖π‘₯ 𝑑π‘₯γ€— =(π‘βˆ’π‘Ž) (π‘™π‘–π‘š)┬(π‘›β†’βˆž) 1/𝑛 (𝑓(π‘Ž)+𝑓(π‘Ž+β„Ž)+𝑓(π‘Ž+2β„Ž)…+𝑓(π‘Ž+(π‘›βˆ’1)β„Ž)) Hence we can write ∫_0^2β–’(π‘₯^2+1) 𝑑π‘₯ =(2βˆ’0) lim┬(nβ†’βˆž) 1/𝑛 (𝑓(0)+𝑓(0+β„Ž)+𝑓(0+2β„Ž)+… +𝑓(0+(π‘›βˆ’1)β„Ž) =2 lim┬(nβ†’βˆž) 1/𝑛 (𝑓(0)+𝑓(β„Ž)+𝑓(2β„Ž)……+𝑓((π‘›βˆ’1)β„Ž) Here, 𝑓(π‘₯)=π‘₯^2+1 𝑓(0)=0^2+1=0+1=1 𝑓(β„Ž)=β„Ž^2+1=(2/𝑛)^2+1=4/𝑛^2 +1 𝑓(2β„Ž)=(2β„Ž)^2+1=γ€–4β„Žγ€—^2+1=4(2/𝑛)^2+1=16/𝑛^2 +1 ….. 𝑓(π‘›βˆ’1)β„Ž=((π‘›βˆ’1)β„Ž)^2+1=γ€–(π‘›βˆ’1)^2 (2/𝑛)γ€—^2+1 =(π‘›βˆ’1)^2 Γ— 4/𝑛^2 +1 Hence, our equation becomes = 2 lim┬(nβ†’βˆž) 1/𝑛 (𝑓(0)+𝑓(β„Ž)+𝑓(2β„Ž)……+𝑓(π‘›βˆ’1)β„Ž) = 2 lim┬(nβ†’βˆž) 1/𝑛 (1+(4/𝑛^2 +1)+(16/𝑛^2 +1" " )+ ……+((4(𝑛 βˆ’ 1)^2)/𝑛^2 +1)) = 2 lim┬(nβ†’βˆž) 1/𝑛 ((1 + 1 + 1…𝑛 π‘‘π‘–π‘šπ‘’π‘ )+0+ 4/𝑛^2 +16/𝑛^2 + …(4(𝑛 βˆ’ 1)^2)/𝑛^2 ) = 2 lim┬(nβ†’βˆž) 1/𝑛 (𝑛 +0+ 4/𝑛^2 +16/𝑛^2 + ……(4(𝑛 βˆ’ 1)^2)/𝑛^2 ) = 2 lim┬(nβ†’βˆž) 1/𝑛 (𝑛+ 4/𝑛^2 (1+4+ ……+(𝑛 βˆ’ 1)^2 ) ) = 2 lim┬(nβ†’βˆž) 1/𝑛 (𝑛+ 4/𝑛^2 (1^2+2^2+ ………+(𝑛 βˆ’ 1)^2 ) ) = 2 lim┬(nβ†’βˆž) 1/𝑛 (𝑛+ 4/𝑛^2 ((𝑛 βˆ’ 1) 𝑛(2𝑛 βˆ’ 1))/6) = 2 lim┬(nβ†’βˆž) 1/𝑛 (𝑛+ 4/𝑛 ((𝑛 βˆ’ 1) (2𝑛 βˆ’ 1))/6) = 2 lim┬(nβ†’βˆž) 1/𝑛 (𝑛+ 2/3𝑛 (π‘›βˆ’1) (2π‘›βˆ’1)) = 2 lim┬(nβ†’βˆž) (𝑛/𝑛 + 2/(3𝑛^2 ) (π‘›βˆ’1) (2π‘›βˆ’1)) We know that 1^2+2^2+ ……+𝑛^2= (𝑛(𝑛 + 1) (2𝑛 +1))/6 1^2+2^2+ ……+(π‘›βˆ’1)^2= ((𝑛 βˆ’ 1)(𝑛 βˆ’ 1+ 1) (2(𝑛 βˆ’1)+1))/6 = ((𝑛 βˆ’ 1) 𝑛(2𝑛 βˆ’ 1))/6 = 2 lim┬(nβ†’βˆž) (𝑛/𝑛 + 2/3 ((𝑛 βˆ’ 1))/𝑛 ((2𝑛 βˆ’ 1))/𝑛) = 2 lim┬(nβ†’βˆž) (1+ 2/3 (1βˆ’ 1/𝑛) (2βˆ’ 1/𝑛)) = 2 (1+ 2/3 (1βˆ’0) (2βˆ’0)) = 2 (1+ 2/3 Γ—2) = 2 (1+ 4/3) = 2 Γ— 7/3 = πŸπŸ’/πŸ‘ (lim┬(nβ†’βˆž) 1/𝑛=0" " )

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.