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 Example 9 - Using integration find area bounded by triangle - Area between curve and curve

Example 9 - Chapter 8 Class 12 Application of Integrals - Part 2
Example 9 - Chapter 8 Class 12 Application of Integrals - Part 3 Example 9 - Chapter 8 Class 12 Application of Integrals - Part 4 Example 9 - Chapter 8 Class 12 Application of Integrals - Part 5 Example 9 - Chapter 8 Class 12 Application of Integrals - Part 6 Example 9 - Chapter 8 Class 12 Application of Integrals - Part 7


Transcript

Example 9 Using integration find the area of region bounded by the triangle whose vertices are (1, 0), (2, 2) and (3, 1) Area of ∆ formed by point 1 , 0﷯ , 2 ,2﷯ & 3 , 1﷯ Step 1: Draw the figure Area ABD Area ABD= 1﷮2﷮𝑦 𝑑𝑥﷯ 𝑦→ equation of line AB Equation of line between A(1, 0) & B(2, 2) is 𝑦 − 0﷮𝑥 − 1﷯= 2 − 0﷮2 − 1﷯ 𝑦﷮𝑥 − 1﷯= 2﷮1﷯ y = 2(x – 1) y = 2x – 2 Area ABD = 1﷮2﷮𝑦 𝑑𝑥﷯ = 1﷮2﷮2 𝑥−1﷯ 𝑑𝑥﷯ = 2 𝑥﷮2﷯﷮2﷯−𝑥﷯﷮1﷮2﷯ =2 2﷮2﷯﷮2﷯−2− 1﷮2﷯﷮2﷯−1﷯﷯ =2 2−2− 1﷮2﷯+1﷯ =2 1﷮2﷯﷯ = 1 Area BDEC Area BDEC = 2﷮3﷮𝑦 𝑑𝑥﷯ 𝑦→ equation of line BC Equation of line between B(2, 2) & C(3, 1) is 𝑦 − 2﷮𝑥 − 2﷯= 1 − 2﷮3 − 2﷯ 𝑦 − 2﷮𝑥 − 2﷯= −1﷮1﷯ y – 2 = –1(x – 2) y – 2 = –x + 2 y = 4 – x Area BDEC = 2﷮3﷮𝑦 𝑑𝑥﷯ = 2﷮3﷮ 4−𝑥﷯﷯ 𝑑𝑥 =4 2﷮3﷮𝑑𝑥−﷯ 2﷮3﷮𝑥𝑑𝑥﷯ =4 𝑥﷯﷮2﷮3﷯− 𝑥﷮2﷯﷮2﷯﷯﷮2﷮3﷯ =4 3−2﷯− 1﷮2﷯ 3﷮2﷯− 2﷮2﷯﷯ =4 ×1− 1﷮2﷯ 9−4﷯ =4− 1﷮2﷯ ×5 = 4− 5﷮2﷯ = 8 − 5﷮2﷯ = 3﷮2﷯ Area ACE Area ACE= 1﷮3﷮𝑦 𝑑𝑥﷯ 𝑦→ equation of line AC Equation of line between A(1, 0) & C(3, 1) is 𝑦 − 0﷮𝑥 − 1﷯= 1 − 0﷮3 − 1﷯ 𝑦﷮𝑥 − 1﷯= 1﷮2﷯ y = 1﷮2﷯ (x – 1) Area ACE = 1﷮3﷮𝑦 𝑑𝑥﷯ = 1﷮3﷮ 1﷮2﷯ 𝑥−1﷯ 𝑑𝑥﷯ = 1﷮2﷯ 1﷮3﷮ 𝑥−1﷯ 𝑑𝑥﷯ = 1﷮2﷯ 𝑥﷮2﷯﷮2﷯−𝑥﷯﷮1﷮3﷯ = 1﷮2﷯ 3﷮2﷯﷮2﷯−3− 1﷮2﷯﷮2﷯−1﷯﷯ = 1﷮2﷯ 9﷮2﷯−3− 1﷮2﷯+1﷯ = 1﷮2﷯ 4﷮2﷯﷯ =1 Hence Area Required = Area ABD + Area BDEC – Area ACE = 1 + 3﷮2﷯−1 = 3﷮2﷯

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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 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.