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Misc 11 - Using integration find area bounded by |x| + |y| = 1 - Area between curve and curve

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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 Hence the figure is Since the Curve symmetrical about 𝑥 & 𝑦−𝑎𝑥𝑖𝑠 Required Area = 4 × Area AOB Area AOB Area ABO = 0﷮1﷮𝑦 𝑑𝑥﷯ where 𝑥+𝑦=1 𝑦=1−𝑥 Therefore, Area ABO = 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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