Question

Find the area bounded by the curve y = cos 2x,

x=0, x=π/2
and x-axis.​

Answer 1

$\textbf{Given:}$

$\text{ y=cos\,2x , x=0 , x=\frac{\pi}{2} and x axis}$

$\textbf{To find:}$

$\text{Area bounded by the given curve and the lines}$

$\textbf{Solution:}$

$\text{From the attachment,}$

$\text{Required area}$

$=\int\limits^{\frac{\pi}{4}}_0\,y\;dx+\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{4}}\,(-y)\;dx$

$=\int\limits^{\frac{\pi}{4}}_0\,cos2x\;dx+\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{4}}\,(-cos2x)\;dx$

$=\int\limits^{\frac{\pi}{4}}_0\,cos2x\;dx-\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{4}}\,cos2x\;dx$

$=[\dfrac{sin2x}{2}]^{\frac{\pi}{4}}_0-[\dfrac{sin2x}{2}]^{\frac{\pi}{2}}_{\frac{\pi}{4}}$

$=\dfrac{1}{2}[sin\frac{\pi}{2}-sin0]-\dfrac{1}{2}[sin\pi-sin\frac{\pi}{2}]$

$=\dfrac{1}{2}[1-0]-\dfrac{1}{2}[0-1]$

$=\dfrac{1}{2}+\dfrac{1}{2}$

$=1\;\text{square units}$

$\therefore\textbf{Area bounded by the given curve and the lines is 1 square units}$