Question 11 - 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 11 Using the method of integration find the area of the region bounded by lines: 2π₯ + π¦ = 4, 3π₯β2π¦=6 and π₯β3π¦+5=0
Plotting the 3 lines on the graph
2π₯ + π¦ = 4
3π₯ β 2π¦ = 6
π₯ β 3π¦ + 5 = 0
Find intersecting Points A & B
Point A
Point A is intersection of lines
x β 3y + 5 = 0 & 2x + y = 4
Now,
x β 3y + 5 = 0
x = 3y β 5
Putting x = 3y β 5 in
2x + y = 4
2(3y β 5) + y = 4
6y β 10 + y = 4
7y = 14
y = 2
Putting y = 2 in
x β 3y + 5 = 0
x β 3(2) + 5 = 0
x β 6 + 5 = 0
x = 1
So, point A (1, 2)
Point B
Point B is intersection of lines
x β 3y + 5 = 0 & 3x β 2y = 6
Now,
x β 3y + 5 = 0
x = 3y β 5
Putting x = 3y β 5 in
3x β 2y = 6
3(3y β 5) β 2y = 6
9y β 15 β 2y = 6
7y = 21
y = 3
Putting y = 3 in
x β 3y + 5 = 0
x β 3(3) + 5 = 0
x β 9 + 5 = 0
x = 4
So, point B is (4, 3)
Finding area
Area Required = Area ABED β Area ACD β Area CBE
Area ABED
Area ABED =β«_1^4βγπ¦ ππ₯γ
π¦β Equation of AB
π₯ β 3π¦+5=0
π₯+5=3π¦
(π₯ + 5)/3=π¦
π¦=(π₯ + 5)/3
Therefore,
Area ABED =β«_1^4βγ((π₯+5)/3) ππ₯γ
=1/3 β«_1^4βγ(π₯+5) ππ₯γ
=1/3 [π₯^2/2+5π₯]_1^4
=1/3 [4^2/2+5.4β[1^2/2+5.1]]
=1/3 [8+20β1/2β5]
=1/3 [45/2]
=15/2
Area ACD
Area ACD =β«_1^2βγπ¦ ππ₯γ
π¦β Equation of line AC
2π₯+π¦=4
π¦=4β2π₯
Area ACD =β«_1^2βγ(4β2π₯" " ) ππ₯γ
=[4π₯β(2π₯^2)/2]_1^2
=[4π₯βπ₯^2 ]_1^2
=[4.2β2^2β[4.1β1^2 ]]
=[8β4β4+1]
= 1
Area CBE
Area CBE =β«_2^4βγπ¦ ππ₯γ
π¦β Equation of line BC
3π₯+2π¦=6
3π₯β6=2π¦
(3π₯ β 6)/2=π¦
π¦=(3π₯ β 6)/2
Therefore,
Area CBE =β«_2^4βγ((3π₯ β 6)/2) ππ₯γ
=1/2 β«_2^4βγ(3π₯β6) ππ₯γ
=1/2 [(3π₯^2)/2β6π₯]_2^4
=1/2 [γ3.4γ^2/2β6.4β[γ3.2γ^2/2β6.2]]
=1/2 [24β24β6+12]
=3
Hence
Area Required = Area ABED β Area ACD β Area CBE
=15/2β1β3
=15/2β4
=(15 β 8)/2
=π/π 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!