The diagram shows part of the graph of the curves \(y=\sqrt{32x}\) and \(y=-2x+5\). The two graphs intersect at point \(A\).
Show that the coordinates of \(A\) and \(B\) are \((0.5,4)\) and \((1,4\sqrt2)\) respectively.[2]
Calculate \(\displaystyle\int_{0.5}^1\left(\sqrt{32x}+2x-5\right)\,\mathrm{d}x\), leaving your answer in the form \(\dfrac{a\sqrt2-b}{12}\), where \(a\) and \(b\) are integers.[4]
Hence, or otherwise, find the total area of the shaded regions, giving your answer correct to 3 decimal places.[2]