IDNLearn.com: Your destination for reliable and timely answers to any question. Find in-depth and accurate answers to all your questions from our knowledgeable and dedicated community members.
Check the picture below.
so we're looking for the area enclosed like so in the picture, now, where sin(x) and cos(x) meet or intersect at least on the 1st Quadrant will be at π/4 where both are 1/√2, so we're really looking for
[tex]\displaystyle\int_{\frac{\pi }{4}}^{\frac{\pi }{2}}~~[sin(x)~~ - ~~cos(x)]dx\implies \int_{\frac{\pi }{4}}^{\frac{\pi }{2}}sin(x)dx~~ - ~~\int_{\frac{\pi }{4}}^{\frac{\pi }{2}}cos(x)dx \\\\\\ \left. \cfrac{}{}-cos(x) \right]_{\frac{\pi }{4}}^{\frac{\pi }{2}}~~ - ~~\left. \cfrac{}{}sin(x) \right]_{\frac{\pi }{4}}^{\frac{\pi }{2}}\implies [0]~~ - ~~\left[-\cfrac{1}{\sqrt{2}} \right]~~ + ~~[-1]~~ - ~~\left[ -\cfrac{1}{\sqrt{2}} \right][/tex]
[tex]\cfrac{1}{\sqrt{2}}-1+\cfrac{1}{\sqrt{2}}\implies \cfrac{2}{\sqrt{2}}-1\implies \sqrt{2}-1[/tex]
