# Example 14 - Chapter 8 Class 12 Application of Integrals (Term 2)

Last updated at Dec. 12, 2019 by

Last updated at Dec. 12, 2019 by

Transcript

Example 14 Prove that curves π¦2=4π₯ and π₯2=4π¦ divide the area of the square bounded by π₯=0, π₯=4, π¦=4 and π¦=0 into three equal parts Drawing figure Here, we have parabolas π¦^2=4π₯ π₯^2=4π¦ And, Square made by the lines x = 4, y = 4, x = 0, y = 0 We need to prove that area of square is divided into 3 parts by the curve So, we need to prove Area OPQA = Area OAQB = Area OBQR Area OPQA Area OPQA = β«_0^4βγπ¦ ππ₯γ Here, 4π¦=π₯^2 π¦=π₯^2/4 So, Area OPQA =β«_0^4βγπ₯^2/4 ππ₯γ = 1/4 [π₯^3/3]_0^4 =1/12Γ[4^3β0^3 ] =1/12 Γ [64β0] =64/12=16/3 Area OPQA is the area by curve x2 = 4y in the x-axis from x = 0 to x = 4 Area OBQR Since Area is on π¦βππ₯ππ , we use formula β«1βγπ₯ ππ¦γ Area OBQR = β«_0^4βγπ₯ ππ¦γ Here, π¦^2=4π₯ π₯=π¦^2/4 So, Area OBQR =β«_0^4βγπ¦^2/4 ππ¦γ = 1/4 [π¦^3/3]_0^4=1/12Γ[4^3β0^3 ]=1/12 Γ[64β0]=16/3 Area OBQR is the area by curve y2 = 4x in the y-axis from y = 0 to y = 4 Area OAQB Area OAQB = Area OBQP β Area OAQP Finding Area OBQP Area OBQP =β«_0^4βγπ¦ ππ₯γ Here, π¦^2=4π₯ π¦=Β±β4π₯ As OBQP is in 1st quadrant, value of y is positive β΄ π¦=β4π₯ Area OBQP =β«_0^4βγβ4π₯ ππ₯γ =2β«_0^4βγβπ₯ ππ₯γ =2β«_0^4βγπ₯^(1/2) ππ₯γ =2 Γ [π₯^(1/2+1)/(1/2+1)]_0^4 =2 Γ [π₯^(3/2)/(3/2)]_0^4 =2 Γ 2/3 [π₯^(3/2) ]_0^4 =4/3 [(4)^(3/2)β(0)^(3/2) ] =4/3 [8β0] =32/3 Area OAQP Area OAQP =β«_0^4βγπ¦ ππ₯γ Here, π₯^2=4π¦ π¦=π₯^2/4 Area OAQP =β«_0^4βγπ₯^2/4 ππ₯γ =1/4 [π₯^(2+1)/(2+1)]_0^4 =1/(4 Γ3) [π₯^3 ]_0^4 =1/12 [4^3β0^3 ] Area OAQP Area OAQP =β«_0^4βγπ¦ ππ₯γ Here, π₯^2=4π¦ π¦=π₯^2/4 Area OAQP =β«_0^4βγπ₯^2/4 ππ₯γ =1/4 [π₯^(2+1)/(2+1)]_0^4 =1/(4 Γ3) [π₯^3 ]_0^4 =1/12 [4^3β0^3 ] =1/12 Γ [64β0] =16/3 β΄ Area OAQB = Area OBQD β Area OAQP = 32/3β16/3 = 16/3 So, Area OAQB = Area OAQP = Area OBRQ = ππ/π square units Hence Proved.

Examples

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Example 8 Important Deleted for CBSE Board 2022 Exams

Example 9 Deleted for CBSE Board 2022 Exams

Example 10 Important Deleted for CBSE Board 2022 Exams

Example 11

Example 12

Example 13 Important

Example 14 Important Deleted for CBSE Board 2022 Exams You are here

Example 15 Important

Chapter 8 Class 12 Application of Integrals (Term 2)

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.