Ex 8.2, 3 - Chapter 8 Class 12 Application of Integrals

Transcript

Ex 8.2 , 3 Find the area of the region bounded by the curves 𝑦=𝑥2+2, 𝑦=𝑥, 𝑥=0 and 𝑥=3 Step 1: Drawing figure 𝑦=𝑥2+2 𝑦−2= 𝑥﷮2﷯ 𝑥﷮2﷯= 𝑦−2﷯ So, it is a parabola And, 𝑥=𝑦 is a line x = 3 is a line x = 0 is the y-axis Step 2: Finding point of intersection B & C For B Putting 𝑥=3 in 𝑥﷮2﷯= 𝑦−2﷯ 3﷮2﷯= 𝑦−2﷯ 9 = 𝑦−2 𝑦=11 Hence B = 3 , 11﷯ For C Putting 𝑥=3 in 𝑥=𝑦 3=𝑦 𝑦=3 Hence C = 3 , 3﷯ Step 3: Finding Area Area required = Area ABDO – Area OCD Area ABDO Area ABDO = 0﷮3﷮𝑦 𝑑𝑥﷯ 𝑦→ eq. of parabola AB 𝑦= 𝑥﷮2﷯+2 ∴ Area ABDO = 0﷮3﷮𝑦 𝑑𝑥﷯ = 0﷮3﷮ 𝑥﷮2﷯+2﷯ 𝑑𝑥﷯ = 𝑥﷮3﷯﷮3﷯+2𝑥﷯﷮0﷮3﷯ = 3﷮3﷯﷮3﷯+2 ×3− 0﷮3﷯﷮3﷯﷯ = 9+6 = 15 Area OCD Area OCD = 0﷮3﷮𝑦 𝑑𝑥﷯ 𝑦→ equation of line 𝑦=𝑥 ∴ Area OCD = 0﷮3﷮𝑦 𝑑𝑥﷯ = 0﷮3﷮𝑥 𝑑𝑥﷯ = 𝑥﷮2﷯﷮2﷯﷯﷮0﷮3﷯ = 3﷮2﷯﷮2﷯− 0﷮2﷯﷮2﷯﷯ = 9﷮2﷯ Area required = Area ABDO – Area OCD = 15 – 9﷮2﷯ = 21﷮2﷯