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Misc 11 - Using integration find area bounded by |x| + |y| = 1

Misc 11 - Chapter 8 Class 12 Application of Integrals - Part 2
Misc 11 - Chapter 8 Class 12 Application of Integrals - Part 3
Misc 11 - Chapter 8 Class 12 Application of Integrals - Part 4
Misc 11 - Chapter 8 Class 12 Application of Integrals - Part 5

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Transcript

Misc 11 Using the method of integration find the area bounded by the curve |𝑥|+|𝑦|=1 [Hint: The required region is bounded by lines 𝑥+𝑦= 1, 𝑥 –𝑦=1, –𝑥+𝑦 =1 and −𝑥 −𝑦=1 ] We know that "│" 𝑥"│"={█(𝑥, 𝑥≥0@&−𝑥, 𝑥<0)┤ & "│" 𝑦"│"={█(𝑦, 𝑦≥0@&−𝑦, 𝑦<0)┤ So, we can write │𝑥"│+│" 𝑦"│"=1 as {█(█(█( 𝑥+𝑦=1 𝑓𝑜𝑟 𝑥>0 , 𝑦>0@−𝑥+𝑦=1 𝑓𝑜𝑟 𝑥<0 𝑦>0)@█( 𝑥−𝑦 =1 𝑓𝑜𝑟 𝑥>0 , 𝑦<0@−𝑥−𝑦=1 𝑓𝑜𝑟 𝑥<0 𝑦<0)))┤ For 𝒙+𝒚=𝟏 For −𝒙+𝒚=𝟏 For −𝒙−𝒚=𝟏 For 𝒙−𝒚=𝟏 Joining them, we get our diagram Since the Curve symmetrical about 𝑥 & 𝑦−𝑎𝑥𝑖𝑠 Required Area = 4 × Area AOB Area AOB Area AOB = ∫_0^1▒〖𝑦 𝑑𝑥〗 where 𝑥+𝑦=1 𝑦=1−𝑥 Therefore, Area AOB = ∫_0^1▒〖(1−𝑥) 𝑑𝑥〗 = [𝑥−𝑥^2/2]_0^1 =1−〖 1〗^2/2−(0−0^2/2)^2 =1−1/2 =1/2 Hence, Required Area = 4 × Area AOB = 4 × 1/2 = 2 square units

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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.