Question 6 - Chapter 8 Class 12 Application of Integrals (Important Question)

Last updated at May 26, 2023 by Teachoo

Since NCERT Books are changed, we are still changing the name of content in images and videos. It would take some time.

But, we assure you that the question is what you are searching for, and the content is the best -Teachoo Promise. If you have any feedback, please contact us.

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class

Question 6 Find the area of the smaller region bounded by the ellipse π₯^2/π^2 +π¦^2/π^2 =1 & π₯/π + π¦/π = 1
Letβs first draw the figure
π^π/π^π +π^π/π^π =π
is an which is a equation ellipse with x-axis as principal axis
And, π/π + π/π = 1
is a line passing through A (a, 0) and B (0, b)
Required Area
Required Area = Area OACB β Area OAB
Area OACB
Area OACB = β«_0^πβγπ¦ ππ₯γ
π¦ β Equation of ellipse
π₯^2/π^2 +π¦^2/π^2 =1
π¦^2/π^2 =1βπ₯^2/π^2
π¦^2=π^2 [1βπ₯^2/π^2 ]
π¦=Β±β(π^2 [1βπ₯^2/π^2 ] )
π¦=Β± πβ(1βπ₯^2/π^2 )
As OACB is in 1st quadrant,
Value of π¦ will be positive
β΄ π¦=πβ(1βπ₯^2/π^2 )
Now,
Area OACB =β«_0^πβγπβ(1βπ₯^2/π^2 )γ ππ₯
=bβ«_0^πβγβ((π^2 β π₯^2)/π^2 ) ππ₯" " γ
=π/π β«_0^πβγβ(π^2βπ₯^2 ) ππ₯" " γ
=π/π [1/2 π₯ β(π^2βπ₯^2 )+π^2/2 sin^(β1)β‘γπ₯/πγ ]_0^π
=π/π [1/2.πβ(π^2βπ^2 )+π^2/2 sin^(β1)β‘γπ/πγβ(1/2 0β(π^2β0^2 )+π^2/2 sin^(β1)β‘γ0/πγ )]
=π/π [0+π^2/2.γπππγ^(βπ)β‘πβ0β0]
=π/π [0+π^2/2.π /π ]
=π/π Γ π^2/2 " Γ " π/2
=( πππ )/4
=π/π [0+π^2/2.π /π ]
=π/π Γ π^2/2 " Γ " π/2
=( πππ )/4
Area OAB
Area OAB =β«_0^πβγπ¦ ππ₯γ
π¦ β Equation of line
π₯/π+π¦/π=1
π¦/π=1βπ₯/π
π¦=π[1βπ₯/π]
Therefore,
Area OAB =β«_0^πβπ[1βπ₯/π]ππ₯
= γπ[π₯βπ₯^2/2π]γ_0^π
= π[πβπ^2/2πβ[0β0^2/2π]]
= π[πβπ/2β0]
= πΓπ/2
=ππ/2
β΄ Area Required = Area OACB β Area OAB
=( π ππ )/4βππ/2
=ππ/2 [π/2β1]
=ππ/2 [(π β 2)/2]
=ππ/π [π βπ] 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.

Hi, it looks like you're using AdBlock :(

Displaying ads are our only source of revenue. To help Teachoo create more content, and view the ad-free version of Teachooo... please purchase Teachoo Black subscription.

Please login to view more pages. It's free :)

Teachoo gives you a better experience when you're logged in. Please login :)

Solve all your doubts with Teachoo Black!

Teachoo answers all your questions if you are a Black user!