To find the area between the x-axis and the parabolic curve, take the integral of the area in which the curve is above the x-axis. function of the graph is y = 4x - x² We can tell by the function (specifically -x²) that the parabola will point downward. To find the domain in which y>0, let's find the roots (0's) of the function: 0 = 4x - x² 0 = x (4 - x) x = 0 or x = 4 Between x=0 and x=4, the curve is above the x-axis. To find the area of the graph, let's take the integral on this range:
First, take the antiderivative of 4x - x²: 2x² - (1/3) x³ Then, plug x=4 into the anti-derivative, and subtract the anti-derivative at x=0: 2(4)² - (1/3)(4³) - (0 - 0) 32 - 64/3 96/3 - 64/3 = 32/3
