Question

Find the area bounded by the curve $y=\cos x$ between $x=0$ and $x=2\pi$.

Answer 1

Given the curve $y=\cos x$ between
$x=0$ and $x=2\pi$

Area Required
$={\int }_0^{\frac{\pi }{2}}ydx+\left|{\int }_{\frac{\pi }{2}}^{\frac{3\pi }{2}}ydx\right|+{\int }_{\frac{3\pi }{2}}^{2\pi }ydx$

$={\int }_0^{\frac{\pi }{2}}\cos xdx+\left|{}_{\frac{\pi }{2}}^{\frac{3\pi }{2}}\cos xdx\right|+{\int }_{\frac{3\pi }{2}}^{2\pi }\cos xdx$

$={\int }_0^{\frac{\pi }{2}}\cos xdx+\left|2{\int }_{\frac{\pi }{2}}^{\pi }\cos xsx\right|+{\int }^2{\pi }_{\frac{3\pi }{2}}\cos xdx$

$=4{\int }_0^{\frac{\pi }{2}}\cos xdx$

$=4{\left[\sin x\right]}_0^{\frac{\pi }{2}}$

$=4[\sin \frac{\pi }{2}-\sin 0]$

$=4[1-0]=4$

Final answer:

Therefore, required area $=4$ square units.