Question

roughly sketch the region enclosed by the curves y=sinx , y=cosx and the x axis between x =0 and pie /2.also find the area of this region

Answer 1

Cos x ≥ Sin x  during  0 ≤ x ≤ π/4
Cos x ≤ Sin x  for    π/4  ≤ x ≤ π/2

Area shaded :  is split into two regions: one under the Sin x curve and the other under the Cos x curve.

$\int\limits^{\frac{\pi}{4}}_0 {Sin\ x} \, dx + \int\limits^{\frac{\pi}{2}}_{\frac{\pi}{4}} {Cos\ x} \, dx \\\\=[-Cos\ x]_0^\frac{\pi}{4}+[Sin\ x]_{\frac{\pi}{4}}^{\frac{\pi}{2}}\\\\=-\frac{1}{\sqrt2}+1+1-\frac{1}{\sqrt2}=2-\sqrt2$