The area in square units of the region bounded by the curve y = 16 - x^2 and lines x = 0, x = 4 above the X-axis is

Step-by-step solution

The given curve is y = 16 - x^2 , which represents the upper half of the circle x^2 + y^2 = 16 with center (0,0) and radius 4 . The region bounded by the curve, x = 0 , x = 4 , and above the X-axis lies entirely in the first quadrant. The required area is given by the integral: A = ₀⁴ 16 - x^2 dx Using the standard integral formula a^2 - x^2 dx = x 2 a^2 - x^2 + a^2 2 ⁻¹ ( x a ) , we get: A = [ x 2 16 - x^2 + 8 ⁻¹ ( x 4 ) ]₀⁴ A = ( 0 + 8 ⁻¹(1) ) - ( 0 + 8 ⁻¹(0) ) A = 8 2 = 4 sq. units. Answer: 4