Check sibling questions

Using integration, find the area of the region in the first quadrant enclosed by the line x + y = 2, the parabola y2 = x and the x-axis.

This question is similar to Misc 7 Chapter 8 Class 12 - Applications of Integrals

  • Slide43.JPG
    Slide44.JPG
    Slide45.JPG
    Slide46.JPG
    Slide47.JPG
    Slide48.JPG
    Slide49.JPG


Transcript

Question 12 (Choice 1) Using integration, find the area of the region in the first quadrant enclosed by the line x + y = 2, the parabola y2 = x and the x-axis. Let’s first draw the Figure Here, π’šπŸ =𝒙 is a Parabola And, x + y = 2 is a straight line Let A be point of intersection of line and parabola And, Point B is (2, 0) Finding point A Since x + y = 2 y = 2 βˆ’ x Putting y = 2 βˆ’ x in equation of parabola 𝑦^2=π‘₯ (πŸβˆ’π’™)^𝟐=𝒙 4+π‘₯^2βˆ’4π‘₯=π‘₯ π‘₯^2βˆ’4π‘₯βˆ’π‘₯+4=0 𝒙^πŸβˆ’πŸ“π’™+πŸ’=𝟎 π‘₯^2βˆ’4π‘₯βˆ’π‘₯+4=0 π‘₯(π‘₯βˆ’4)βˆ’1(π‘₯βˆ’4)=0 (π‘₯βˆ’4)(π‘₯βˆ’1)=0 So, x = 4, x = 1 Since for point A, x-coordinate will be less than 2 ∴ x = 1 Putting x = 1 in equation of line x + y = 2 1 + y = 2 y = 2 βˆ’ 1 y = 1 So, Coordinates of point A = (1, 1) Finding Area Area Required = Area OAD + Area ADB Area OAD Area OAD = ∫_0^1▒〖𝑦 𝑑π‘₯" " γ€— y β†’ Equation of parabola 𝑦^2 = x 𝑦 = Β± √π‘₯ Since OAD is in 1st quadrant, value of y is positive ∴ π’š = βˆšπ’™ Now, Area OAD = ∫_0^1▒〖𝑦 𝑑π‘₯" " γ€— = ∫_𝟎^πŸβ–’γ€–βˆšπ’™ 𝒅𝒙" " γ€— = [π‘₯^(1/2 + 1)/(1/2 + 1)]_0^1 = [π‘₯^(3/2)/(3/2)]_0^1 = 𝟐/πŸ‘ [𝒙^(πŸ‘/𝟐) ]_𝟎^𝟏 = 2/3 [1^(3/2)βˆ’0^(3/2) ] = 𝟐/πŸ‘ Area ADB Area ADB = ∫1_1^2▒〖𝑦 𝑑π‘₯γ€— y β†’ Equation of line x + y = 2 y = 2 βˆ’ x Therefore, Area ADB = ∫1_𝟏^πŸβ–’(πŸβˆ’π’™)𝒅𝒙 = [2π‘₯βˆ’π‘₯^2/2]_1^2 = [(2(2)βˆ’2^2/2)βˆ’(2(1)βˆ’1^2/2)] = [2π‘₯βˆ’π‘₯^2/2]_1^2 = [(2(2)βˆ’2^2/2)βˆ’(2(1)βˆ’1^2/2)] = [(4βˆ’2)βˆ’(2βˆ’1/2)] = 2βˆ’3/2 = 𝟏/𝟐 Thus, Area Required = Area OAD + Area ADB = 2/3+1/2 = πŸ•/πŸ” square units

Davneet Singh's photo - Teacher, Engineer, Marketer

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.