The area bounded by the curve y = x | ( x 2 ) | and the X-axis between the lines x = 0 and x = 4 (in square units), is....

Step-by-step solution

The required area is given by the integral: A = ₀^ 4 x | ( x 2 ) | dx Since ( x 2 ) 0 for x [0, 2 ] and ( x 2 ) 0 for x [2 , 4 ] , the integral can be split as: A = ₀^ 2 x ( x 2 ) dx - _ 2 ^ 4 x ( x 2 ) dx Using integration by parts: x ( x 2 ) dx = x (-2 ( x 2 ) ) - 1 (-2 ( x 2 ) ) dx = -2x ( x 2 ) + 4 ( x 2 ) Evaluating the first integral: ₀^ 2 x ( x 2 ) dx = [-2x ( x 2 ) + 4 ( x 2 ) ]₀^ 2 = (-4 (-1) + 0) - (0 + 0) = 4 Evaluating the second integral: _ 2 ^ 4 x ( x 2 ) dx = [-2x ( x 2 ) + 4 ( x 2 ) ]_ 2 ^ 4 = (-8 (