Ex 8.2, 4

Transcript

Ex 8.2 , 4 Using integration find the area of region bounded by the triangle whose vertices are (– 1, 0), (1, 3) and (3, 2) Area of ∆ formed by points A(– 1, 0), B(1, 3) and C(3, 2) Step 1: Draw the figure Area ABD Area ABD= −1﷮1﷮𝑦 𝑑𝑥﷯ 𝑦→ equation of line AB Equation of line between A(–1, 0) & B(1, 3) is 𝑦 − 0﷮𝑥 − (−1)﷯= 3 − 0﷮1 − (−1)﷯ 𝑦﷮𝑥 + 1﷯= 3﷮2﷯ y = 3﷮2﷯ (x + 1) Area ABD = −1﷮1﷮𝑦 𝑑𝑥﷯ = −1﷮1﷮ 3﷮2﷯ 𝑥+1﷯﷯ dx = 3﷮2﷯ −1﷮1﷮ 𝑥+1﷯﷯ dx = 3﷮2﷯ 𝑥﷮2﷯﷮2﷯+𝑥﷯﷮−1﷮1﷯ = 3﷮2﷯ 1﷮2﷯﷮2﷯+1﷯− −1﷯﷮2﷯﷮2﷯+(−1)﷯﷯ = 3﷮2﷯ 3﷮2﷯﷯− −1﷮2﷯﷯﷯ = 3﷮2﷯ × 2 = 3 Area BDEC Area BDEC = 1﷮3﷮𝑦 𝑑𝑥﷯ 𝑦→ equation of line BC Equation of line between B(1, 3) & C(3, 2) is 𝑦 − 3﷮𝑥 − 1﷯= 2 − 3﷮3 − 1﷯ 𝑦 − 3﷮𝑥 − 1﷯= −1﷮2﷯ 2(y – 3) = –1(x – 1) 2y – 6 = –x + 1 2y = – x + 7 y = 1﷮2﷯ (–x + 7) Area BDEC = 1﷮3﷮𝑦 𝑑𝑥﷯ = 1﷮3﷮ 1﷮2﷯ −𝑥+7﷯𝑑﷯x = 1﷮2﷯ −𝑥﷮2﷯﷮2﷯+7𝑥﷯﷮1﷮3﷯ = 1﷮2﷯ −3﷮2﷯﷮2﷯+7(3)﷯− −1﷮2﷯﷮2﷯+7(1)﷯﷯ = 1﷮2﷯ −9﷮2﷯+21+ 1﷮2﷯−7﷯ = 1﷮2﷯ −9 + 1﷮2﷯+14﷯ = 1﷮2﷯ −8﷮2﷯+14﷯ = 1﷮2﷯ −4+14﷯ = 10﷮2﷯ = 5 Area ACE Area ACE= −1﷮3﷮𝑦 𝑑𝑥﷯ 𝑦→ equation of line AC Equation of line between A(–1, 0) & C(3, 2) is 𝑦 − 0﷮𝑥 − (−1)﷯= 2 − 0﷮3 − (−1)﷯ 𝑦﷮𝑥 + 1﷯= 2﷮4﷯ y = 1﷮2﷯ (x + 1) Area ACE = −1﷮3﷮𝑦 𝑑𝑥﷯ = −1﷮3﷮ 1﷮2﷯ 𝑥+1﷯𝑑﷯x = 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﷯ 9﷮2﷯+ 1﷮2﷯+3+1﷯ = 1﷮2﷯ 4+4﷯ = 4 Hence, Required area = Area ABD + Area BDEC − Area ACE = 3 + 5 − 4 = 4 units