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Ex 7.8, 3 - Integrate (x2 dx) from 2 to 3 by limit as a sum  - Ex 7.8

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Ex7.8, 3 ﷐2﷮3﷮𝑥2 𝑑𝑥﷯ ﷐2﷮3﷮𝑥2 𝑑𝑥﷯ Putting 𝑎 =2 𝑏 =3 ℎ=﷐𝑏 − 𝑎﷮𝑛﷯ =﷐3 − 2﷮𝑛﷯ =﷐1﷮𝑛﷯ 𝑓﷐𝑥﷯=﷐𝑥﷮2﷯ Hence we can write it as ﷐2﷮3﷮𝑥2 𝑑𝑥﷯ =﷐3−2﷯﷐𝑙𝑖𝑚﷮𝑛→𕔴uc1﷯﷐1﷮𝑛﷯﷐𝑓﷐2﷯+𝑓﷐2+ℎ﷯+𝑓﷐2+2ℎ﷯+ …+𝑓﷐2+﷐𝑛−1﷯ℎ﷯﷯ 𝑓﷐𝑥﷯=﷐𝑥﷮2﷯ 𝑓﷐2﷯=﷐﷐2﷯﷮2﷯=4 𝑓﷐2+ℎ﷯=﷐﷐2+ℎ﷯﷮2﷯ 𝑓 ﷐2+2ℎ﷯=﷐﷐2+2ℎ﷯﷮2﷯ … 𝑓﷐2+﷐𝑛−1﷯ℎ﷯=﷐﷐2+﷐𝑛−1﷯ℎ﷯﷮2﷯ Thus, ﷐2﷮3﷮𝑥2 𝑑𝑥﷯ =﷐3−2﷯﷐𝑙𝑖𝑚﷮𝑛→𕔴uc1﷯﷐1﷮𝑛﷯﷐𝑓﷐2﷯+𝑓﷐2+ℎ﷯+𝑓﷐2+2ℎ﷯+ …+𝑓﷐2+﷐𝑛−1﷯ℎ﷯﷯ =1 ﷐𝑙𝑖𝑚﷮𝑛→𕔴uc1﷯﷐1﷮𝑛﷯﷐﷐﷐2﷯﷮2﷯+﷐﷐2+ℎ﷯﷮2﷯+﷐﷐2+2ℎ﷯﷮2﷯+ …+﷐﷐2+﷐𝑛−1﷯ℎ﷯﷮2﷯﷯ =﷐𝑙𝑖𝑚﷮𝑛→𕔴uc1﷯﷐1﷮𝑛﷯﷐﷐﷐2﷯﷮2﷯+﷐﷐2﷯﷮2﷯+﷐﷐ℎ﷯﷮2﷯+2﷐2﷯﷐ℎ﷯+﷐﷐2﷯﷮2﷯ ﷐﷐2ℎ﷯﷮2﷯+2﷐2﷯﷐2ℎ﷯ + …﷐﷐2﷯﷮2﷯+﷐﷐﷐𝑛−1﷯ℎ﷯﷮2﷯+2×2×﷐𝑛−1﷯ ℎ﷯ =﷐𝑙𝑖𝑚﷮𝑛→𕔴uc1﷯﷐1﷮𝑛﷯ ﷐﷐2﷮2﷯+﷐2﷮2﷯+ … +﷐﷐2﷯﷮2﷯﷯ +﷐𝑛﷮2﷯+﷐﷐2ℎ﷯﷮2﷯+ … +﷐﷐𝑛−1﷯ℎ﷮2﷯ +﷐2﷐2﷯﷐ℎ﷯+2﷐2﷯2ℎ+ … +2﷐2﷯﷐𝑛−1﷯ℎ﷯ =﷐𝑙𝑖𝑚﷮𝑛→𕔴uc1﷯﷐1﷮𝑛﷯ (﷐𝑛﷐2﷯﷮2﷯+﷐﷐ℎ﷮2﷯+﷐﷐2﷯﷮2﷯ . ﷐ℎ﷮2﷯+ … +﷐﷐𝑛−1﷯﷮2﷯﷐ℎ﷮2﷯﷯ +﷐2﷐2﷯﷐ℎ﷯+22﷐2﷯﷐ℎ﷯+ …+2﷐2﷯﷐𝑛−1﷯ℎ﷯ ) =﷐𝑙𝑖𝑚﷮𝑛→𕔴uc1﷯﷐1﷮𝑛﷯(𝑛﷐﷐2﷯﷮2﷯+﷐ℎ﷮2﷯﷐﷐﷐1﷯﷮2﷯+﷐﷐2﷯﷮2﷯+ …+﷐﷐𝑛−1﷯﷮2﷯﷯ + 4ℎ ﷐1+2+ …+﷐𝑛−1﷯﷯) =﷐𝑙𝑖𝑚﷮𝑛→𕔴uc1﷯﷐1﷮𝑛﷯﷐𝑛﷐﷐2﷯﷮2﷯+﷐ℎ﷮2﷯﷐﷐𝑛﷐𝑛 − 1﷯﷐2𝑛 − 1﷯﷮6﷯﷯+4ℎ﷐﷐𝑛﷐𝑛 − 1﷯﷮2﷯﷯ ﷯ =﷐𝑙𝑖𝑚﷮𝑛→𕔴uc1﷯﷐1﷮𝑛﷯﷐4𝑛+﷐ℎ﷮2﷯ ﷐﷐𝑛﷐𝑛 − 1﷯﷐2𝑛 − 1﷯﷯﷮6﷯+2ℎ﷐𝑛﷐𝑛 − 1﷯﷯ ﷯ =﷐𝑙𝑖𝑚﷮𝑛→𕔴uc1﷯﷐﷐4𝑛﷮𝑛﷯+﷐ℎ﷮2﷯ ﷐﷐𝑛﷐𝑛 − 1﷯﷐2𝑛 − 1﷯﷮6𝑛﷯﷯+2ℎ﷐﷐𝑛﷐𝑛 − 1﷯﷮𝑛﷯﷯﷯ =﷐𝑙𝑖𝑚﷮𝑛→𕔴uc1﷯﷐4+﷐ℎ﷮2﷯﷐﷐﷐𝑛 − 1﷯﷐2𝑛 − 1﷯﷮6﷯﷯+2ℎ﷐﷐𝑛 − 1﷯﷯﷯ =﷐𝑙𝑖𝑚﷮𝑛→𕔴uc1﷯﷐4+﷐﷐﷐1﷮𝑛﷯﷯﷮2﷯﷐﷐𝑛 − 1﷯﷐2𝑛 − 1﷯﷮6﷯+2﷐﷐1﷮𝑛﷯﷯﷐𝑛 − 1﷯﷯ =﷐𝑙𝑖𝑚﷮𝑛→𕔴uc1﷯﷐4+﷐1﷮﷐𝑛﷮2﷯﷯ . ﷐﷐𝑛 − 1﷯﷐2𝑛 − 1﷯﷮6﷯ +2﷐1 − ﷐1﷮𝑛﷯﷯﷯ =﷐𝑙𝑖𝑚﷮𝑛→𕔴uc1﷯﷐4+ ﷐﷐1 − ﷐1﷮𝑛﷯﷯﷐2 − ﷐1﷮𝑛﷯﷯﷮6﷯ +2﷐1 − ﷐1﷮𝑛﷯﷯﷯ =4+ ﷐﷐1 − ﷐1﷮𕔴uc1﷯﷯﷐2 − ﷐1﷮𕔴uc1﷯﷯﷮6﷯ +2﷐1 − ﷐1﷮𕔴uc1﷯﷯ =4+ ﷐﷐1 − 0﷯﷐2 − 0﷯﷮6﷯ +2﷐1 −0﷯ =4+ ﷐1 . 2﷮6﷯ +2 .1 =4+ ﷐1﷮3﷯ +2 =6+ ﷐1﷮3﷯ = ﷐𝟏𝟗﷮𝟑﷯

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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