Question Number 66502 by mhmd last updated on 16/Aug/19
$${find}\:{the}\:{area}\:{about}\:{cos}\left(\mathrm{2}\theta\right) \\ $$
Answered by mr W last updated on 16/Aug/19
![A=8∫_0 ^(π/4) ((r^2 dθ)/2) =4∫_0 ^(π/4) cos^2 2θ dθ =2∫_0 ^(π/4) (1+cos 4θ) dθ =2[θ+((sin 4θ)/4)]_0 ^(π/4) =(π/2)](Q66541.png)
$${A}=\mathrm{8}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \frac{{r}^{\mathrm{2}} {d}\theta}{\mathrm{2}} \\ $$$$=\mathrm{4}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \mathrm{cos}^{\mathrm{2}} \:\mathrm{2}\theta\:{d}\theta \\ $$$$=\mathrm{2}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \left(\mathrm{1}+\mathrm{cos}\:\mathrm{4}\theta\right)\:{d}\theta \\ $$$$=\mathrm{2}\left[\theta+\frac{\mathrm{sin}\:\mathrm{4}\theta}{\mathrm{4}}\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \\ $$$$=\frac{\pi}{\mathrm{2}} \\ $$