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}`