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

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Question 8 Using the method of integration find the area bounded by the curve |π₯|+|π¦|=1 [Hint: The required region is bounded by lines π₯+π¦= 1, π₯ βπ¦=1, βπ₯+π¦ =1 and βπ₯ βπ¦=1 ]
We know that
"β" π₯"β"={β(π₯, π₯β₯0@&βπ₯, π₯<0)β€ & "β" π¦"β"={β(π¦, π¦β₯0@&βπ¦, π¦<0)β€
So, we can write βπ₯"β+β" π¦"β"=1
as {β(β(β( π₯+π¦=1 πππ π₯>0 , π¦>0@βπ₯+π¦=1 πππ π₯<0 π¦>0)@β( π₯βπ¦ =1 πππ π₯>0 , π¦<0@βπ₯βπ¦=1 πππ π₯<0 π¦<0)))β€
For π+π=π
For βπ+π=π
For βπβπ=π
For πβπ=π
Joining them,
we get our diagram
Since the Curve symmetrical about π₯ & π¦βππ₯ππ
Required Area = 4 Γ Area AOB
Area AOB
Area AOB = β«_0^1βγπ¦ ππ₯γ
where
π₯+π¦=1
π¦=1βπ₯
Therefore,
Area AOB = β«_0^1βγ(1βπ₯) ππ₯γ
= [π₯βπ₯^2/2]_0^1
=1βγ 1γ^2/2β(0β0^2/2)^2
=1β1/2
=1/2
Hence,
Required Area = 4 Γ Area AOB
= 4 Γ 1/2
= 2 square units

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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