Question

Find the area bounded by the curve
y = Cos x between x = 0 and x = 2π.

Please Show the graph.

Answer 1

Answer = 4 sq units.

See the diagram enclosed .. of Cos x from 0 to 2π.

We need the area under the curve (enclosed between X-axis and the curve).

Bounded area = 
$A=\int \limits_{x=0}^{2\pi} {| \: Cos \: x \: |} \, dx\\\\\A=\int \limits_{x=0}^{\pi/2} {| \: Cos \: x \: |} \, dx + \int \limits_{x=\pi/2}^{\pi} {| \: Cos \: x \: |} \, dx \\\\\+\int \limits_{x=\pi}^{3\pi/2} {| \: Cos \: x \: |} \, dx + \int \limits_{x=3\pi/2}^{2\pi} {| \: Cos \: x \: |} \, dx\\\\=4 * \int \limits_{x=0}^{\pi/2} {| \: Cos \: x \: |} \, dx \\\\=4* [ Sin x ]_0^{\pi/2}=4*[sin \frac{\pi}{2}-sin0]\\\\=4*1=4$

So Answer = 4 
  Bounded Area = Area A1 + Area A2 + Area A3 + Area A4.

  From symmetry of the curve we know that all the above four areas are equal.
We can also prove that by taking integrals for each of these parts.