Question

The area of the region between the x- axis and the curve y=cosx when x lies between zero and 2π.

Answer 1

To find area under a curve, integrate it.
If you integrate cos x with respect to x, you get sin(x)
If you directly apply the limits from 0 to 2pi, you will get 0.
To find the area, you need to apply limits from 0 to π/2, π/2 to π, 3π/2 and 3π/2 to 2π. Take modulus of each limit and add.
You will get the answer as 4. (value of each limit 1)
 

Answer 2

I hope you know integration.

You construct small rectangles of height Cos x and width dx along x-axis from x = 0 to x = 2 π.  Add the areas of these rectangles to get the answer. Now, from x = 0 to x = π/2, Cos x is positive. Let us take this area and multiply with 4 to get the area from x = 0 to x = 2π.

$\int\limits^{2 \pi }_{0 } {f(x)} \, dx = Area\ bounded\ between\ x-axis\ and\ curve\ f(x).$

$Area = 4 \int\limits^{ \pi /2}_0 {Cos\ x} \, dx = 4\ [ Sin\ x ]^\frac{ \pi }{2}_{0} = 4 [ 1 - 0 ] \\ \\ =\ 4\ square\ units$