Misc 14 - Find area bounded by lines: 2x + y = 4, 3x-2y=6 - Miscellaneous

Slide2.JPG
Slide3.JPG Slide4.JPG Slide5.JPG Slide6.JPG Slide7.JPG Slide8.JPG Slide9.JPG Slide10.JPG

  1. Class 12
  2. Important Question for exams Class 12
Ask Download

Transcript

Misc 14 Using the method of integration find the area of the region bounded by lines: 2𝑥 + 𝑦 = 4, 3𝑥–2𝑦=6 and 𝑥–3𝑦+5=0 Step 1: Make Figure 2𝑥 + 𝑦 = 4 3𝑥 – 2𝑦 = 6 𝑥 – 3𝑦 + 5 = 0 Step 2: Find intersecting Points A & B Point A Point A is intersection of lines x – 3y + 5 = 0 & 2x + y = 4 Now, x – 3y + 5 = 0 x = 3y – 5 Putting x = 3y – 5 in 2x + y = 4 2(3y – 5) + y = 4 6y – 10 + y = 4 7y = 14 y = 2 Putting y = 2 in x – 3y + 5 = 0 x – 3(2) + 5 = 0 x – 6 + 5 = 0 x = 1 So, point A (1, 2) Point B Point B is intersection of lines x – 3y + 5 = 0 & 3x – 2y = 6 Now, x – 3y + 5 = 0 x = 3y – 5 Putting x = 3y – 5 in 3x – 2y = 6 3(3y – 5) – 2y = 6 9y – 15 – 2y = 6 7y = 21 y = 3 Putting y = 3 in x – 3y + 5 = 0 x – 3(3) + 5 = 0 x – 9 + 5 = 0 x = 4 So, point A (4, 3) Finding area Area ABED Area ABED = 1﷮4﷮𝑦 𝑑𝑥﷯ 𝑦→ Equation of AB 𝑥 – 3𝑦+5=0 𝑥+5=3𝑦 𝑥 + 5﷮3﷯=𝑦 𝑦= 𝑥 + 5﷮3﷯ Therefore, Area ABED = 1﷮4﷮ 𝑥+5﷮3﷯﷯ 𝑑𝑥﷯ = 1﷮3﷯ 1﷮4﷮ 𝑥+5﷯ 𝑑𝑥﷯ = 1﷮3﷯ 𝑥﷮2﷯﷮2﷯+5𝑥﷯﷮1﷮4﷯ = 1﷮3﷯ 4﷮2﷯﷮2﷯+5.4− 1﷮2﷯﷮2﷯+5.1﷯﷯ = 1﷮3﷯ 8+20− 1﷮2﷯−5﷯ = 1﷮3﷯ 45﷮2﷯﷯ = 15﷮2﷯ Area ACD Area ACD = 1﷮2﷮𝑦 𝑑𝑥﷯ 𝑦→ Equation of line AC 2𝑥+𝑦=4 𝑦=4−2𝑥 Area ACD = 1﷮2﷮ 4−2𝑥 ﷯ 𝑑𝑥﷯ = 4𝑥− 2 𝑥﷮2﷯﷮2﷯﷯﷮1﷮2﷯ = 4𝑥− 𝑥﷮2﷯﷯﷮1﷮2﷯ = 4.2− 2﷮2﷯− 4.1− 1﷮2﷯﷯﷯ = 8−4−4+1﷯ = 1 Area CBE Area CBE = 2﷮4﷮𝑦 𝑑𝑥﷯ 𝑦→ Equation of line BC 3𝑥+2𝑦=6 3𝑥−6=2𝑦 3𝑥 − 6﷮2﷯=𝑦 𝑦= 3𝑥 − 6﷮2﷯ Therefore, Area CBE = 2﷮4﷮ 3𝑥 − 6﷮2﷯﷯ 𝑑𝑥﷯ = 1﷮2﷯ 2﷮4﷮ 3𝑥−6﷯ 𝑑𝑥﷯ = 1﷮2﷯ 3 𝑥﷮2﷯﷮2﷯−6𝑥﷯﷮2﷮4﷯ = 1﷮2﷯ 3.4﷮2﷯﷮2﷯−6.4− 3.2﷮2﷯﷮2﷯−6.2﷯﷯ = 1﷮2﷯ 24−24−6+12﷯ =3 Hence Area Required = Area ABED – Area ACD – Area CBE = 15﷮2﷯−1−3 = 15﷮2﷯−4 = 15 − 8﷮2﷯ = 7﷮2﷯ square units

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 7 years. He provides courses for Mathematics and Science from Class 6 to 12. You can learn personally from here https://www.teachoo.com/premium/maths-and-science-classes/.
Jail