Question

Graph the curves \(y = {x^2} - x\) and \(y = {x^3} - 4{x^2} + 3x\) on a common screen and observe that the region between them consists of two parts. Find the area of this region.

Short answer

The area of the region is \(\frac{7}{{12}}\).

Step-by-step solution

The area \(A\) of the region bounded by the curves is given by

The area \(A\) of the region bounded by the curves \(y = f(x),y = g(x)\), and the lines \(x = a,x = b\), where \(f\) and \(g\) are continuous and \(f(x) \ge g(x)\) for all \(x\) in (a, b), is \(A = \int_a^b {(f(x) - g(x))dx} \)

Sketch the graph of the curves

Sketch the graph of the curve\(y = {x^2} - x\) and \(y = {x^3} - 4{x^2} + 3x\) as shown in the figure below