Question 7 - Chapter 8 Class 12 Application of Integrals (Important Question)

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Question 7
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= 12𝑦 𝑑𝑥
𝑦→ equation of line AB
Equation of line between A(1, 0) & B(2, 2) is
𝑦 − 0𝑥 − 1= 2 − 02 − 1
𝑦𝑥 − 1= 21
y = 2(x – 1)
y = 2x – 2
Area ABD = 12𝑦 𝑑𝑥
= 122 𝑥−1 𝑑𝑥
= 2 𝑥22−𝑥12
=2 222−2− 122−1
=2 2−2− 12+1
=2 12
= 1
Area BDEC
Area BDEC = 23𝑦 𝑑𝑥
𝑦→ equation of line BC
Equation of line between B(2, 2) & C(3, 1) is
𝑦 − 2𝑥 − 2= 1 − 23 − 2
𝑦 − 2𝑥 − 2= −11
y – 2 = –1(x – 2)
y – 2 = –x + 2
y = 4 – x
Area BDEC = 23𝑦 𝑑𝑥
= 23 4−𝑥 𝑑𝑥
=4 23𝑑𝑥− 23𝑥𝑑𝑥
=4 𝑥23− 𝑥2223
=4 3−2− 12 32− 22
=4 ×1− 12 9−4
=4− 12 ×5
= 4− 52
= 8 − 52
= 32
Area ACE
Area ACE= 13𝑦 𝑑𝑥
𝑦→ equation of line AC
Equation of line between A(1, 0) & C(3, 1) is
𝑦 − 0𝑥 − 1= 1 − 03 − 1
𝑦𝑥 − 1= 12
y = 12 (x – 1)
Area ACE = 13𝑦 𝑑𝑥
= 13 12 𝑥−1 𝑑𝑥
= 12 13 𝑥−1 𝑑𝑥
= 12 𝑥22−𝑥13
= 12 322−3− 122−1
= 12 92−3− 12+1
= 12 42
=1
Hence
Area Required = Area ABD + Area BDEC – Area ACE
= 1 + 32−1
= 32

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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