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Ex 7.11, 18 Class 12 - Evaluate definite integral |x - 1| from 0 to 4

Ex 7.11, 18 - Chapter 7 Class 12 Integrals - Part 2


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Ex 7.11, 18 By using the properties of definite integrals, evaluate the integrals : ∫_0^4β–’|π‘₯βˆ’1| 𝑑π‘₯ |π‘₯βˆ’1|= {β–ˆ( (π‘₯βˆ’1) 𝑖𝑓 π‘₯βˆ’1β‰₯[email protected]βˆ’(π‘₯βˆ’1) 𝑖𝑓 π‘₯βˆ’1<0)─ = {β–ˆ((π‘₯βˆ’1,) 𝑖𝑓 π‘₯β‰₯[email protected]βˆ’(π‘₯βˆ’1) 𝑖𝑓 π‘₯<1)─ ∴ ∫_0^4β–’|π‘₯βˆ’1|𝑑π‘₯=∫_0^1β–’|π‘₯βˆ’1|𝑑π‘₯+∫_1^4β–’|π‘₯βˆ’1|𝑑π‘₯ Using the property, P2 P2 :- ∫_π‘Ž^𝑏▒〖𝑓(π‘₯)𝑑π‘₯=γ€— ∫_π‘Ž^𝑐▒〖𝑓(π‘₯)𝑑π‘₯+∫_𝑐^𝑏▒𝑓(π‘₯)𝑑π‘₯γ€— =∫_0^1β–’γ€–βˆ’(π‘₯βˆ’1)𝑑π‘₯+γ€— ∫_1^4β–’(π‘₯βˆ’1)𝑑π‘₯ =∫_0^1β–’γ€–(βˆ’π‘₯+1)𝑑π‘₯+γ€— ∫_1^4β–’(π‘₯βˆ’1)𝑑π‘₯ =∫_0^1β–’γ€–βˆ’π‘₯ 𝑑π‘₯+γ€— ∫_0^1β–’γ€–1. 𝑑π‘₯+∫_1^4β–’γ€–π‘₯ . 𝑑π‘₯βˆ’βˆ«_1^4β–’γ€–1.𝑑π‘₯γ€—γ€—γ€— =βˆ’[π‘₯^2/2]_0^1+[π‘₯]_0^1βˆ’[π‘₯^2/2]_1^4βˆ’[π‘₯]_1^4 =βˆ’[((1)^2 βˆ’ 0)/2]+[1βˆ’0]+[((4)^2βˆ’(1)^2)/2]βˆ’[4βˆ’1] =βˆ’1/2+1+[(16 βˆ’ 1)/2]βˆ’3 =βˆ’1/2+15/2βˆ’3+1 =(14 )/2βˆ’2= 7 βˆ’ 2 = 5

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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.