The area of the region bounded by the curve y = x^3 and the lines y = 8 and x = 0 is ... square units.

Step-by-step solution

The required area is bounded by the curve y = x^3 , the line y = 8 , and the y-axis ( x = 0 ). The point of intersection of y = x^3 and y = 8 in the first quadrant is x = 2 . The area can be calculated by integrating with respect to x from x = 0 to x = 2 : A = ₀² (8 - x^3) dx A = [ 8x - x^4 4 ]₀² A = 16 - 16 4 A = 16 - 4 = 12 Alternatively, integrating with respect to y from y = 0 to y = 8 : A = ₀⁸ x dy = ₀⁸ y^ 1/3 dy A = [ 3 4 y^ 4/3 ]₀⁸ A = 3 4 (8^ 4/3 ) = 3 4 (16) = 12 Answer: 12