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Area bounded by curve and horizontal or vertical line
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Ex 8.1, 4 (MCQ) Important
Misc 1 (i)
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Area bounded by curve and horizontal or vertical line
Last updated at May 29, 2023 by Teachoo
Ex 8.1, 12 Area lying in the first quadrant and bounded by the circle 𝑥2+𝑦2=4 and the lines 𝑥 = 0 and 𝑥 = 2 is (A) π (B) 𝜋2 (C) 𝜋3 (D) 𝜋4 Equation of Given Circle :- 𝑥2+ 𝑦2=4 𝑥2+ 𝑦2= 22 ∴ 𝑟𝑎𝑑𝑖𝑢𝑠 , 𝑟=2 Line 𝑥=0 is y-axis & Line x = 2 passes through point A 2 , 0 So, Required area = Area of shaded region = Area OAB = 02𝑦.𝑑𝑥 We know that, 𝑥2+ 𝑦2=4 𝑦2=4− 𝑥2 ∴ 𝑦=± 4− 𝑥2 As, OBA is in 1st Quadrant ∴ 𝑦= 4− 𝑥2 ∴ Required area = 02𝑦.𝑑𝑥 = 02 4− 𝑥2 𝑑𝑥 = 02 22− 𝑥2 𝑑𝑥 = 𝑥 2 22− 𝑥2+2 sin−1 𝑥2 02 = 22 22− 22+2 sin−1 22 − 02 22− 02−2 sin−1 02 = 0+2 sin−1 1−0 4−2 sin−1 0 = 2 sin−1 1−2 sin−1 0−0 = 2 sin−1 1− sin−1 0 = 2 𝜋2−0 = 2 . 𝜋2 = π ∴ Area Required = π square units Hence, Option (A) is correct