Question

find the area under (sin x) bounded by x= 0 and x = 2π and x-axis

Answer 1

You probably want the unsigned area, which means you don't compute the integral

$\displaystyle\int_0^{2\pi}\sin x\,\mathrm dx$

but rather, the integral of the absolute value,

$\displaystyle\int_0^{2\pi}|\sin x|\,\mathrm dx$

$\sin x$ is positive when $0<x<\pi$ and negative when $\pi<x<2\pi$, so

$\displaystyle\int_0^{2\pi}|\sin x|\,\mathrm dx=\int_0^\pi\sin x\,\mathrm dx-\int_\pi^{2\pi}\sin x\,\mathrm dx$

$\displaystyle\int_0^{2\pi}|\sin x|\,\mathrm dx=(-\cos x)\bigg|_0^\pi-(-\cos x)\bigg|_\pi^{2\pi}$

$\displaystyle\int_0^{2\pi}|\sin x|\,\mathrm dx=\boxed{4}$