The area of the region (in sq.units) bounded by the curve y = 2 1 - x^2 and the X-axis is _____

Step-by-step solution

The given curve is y = 2 1 - x^2 . Squaring both sides, we get y^2 = 4(1 - x^2) x^2 + y^2 4 = 1 . This represents an ellipse with semi-minor axis a = 1 and semi-major axis b = 2 . Since y 0 , the given curve represents the upper half of the ellipse. The area of the complete ellipse is a b = (1)(2) = 2 . Therefore, the area of the upper half of the ellipse is 1 2 (2 ) = . Alternatively, the area can be found by integration: Area = _ -1 ¹ 2 1 - x^2 dx Area = 2 [ x 2 1 - x^2 + 1 2 ⁻¹x ]_ -1 ¹ Area = 2 [ 1 2 ( 2 ) - 1