1.1 Area between curve and $y$-axis

  Area of Bounded Regions

The area between the curve $x=f(y)$, the $y$-axis and the lines $y=c$ and $y=d$ is given by,

$$A = \int\limits_a^b {f(y)dy} $$

Question 1. Find the area bounded by the curve $y = \operatorname{sinx} $, $x$-axis, $y$-axis and $x = \pi $.

Solution: The required area is given as shown in the figure.

$$A = \int\limits_0^\pi {\sin xdx} $$$$ \Rightarrow A = [ - \operatorname{cosx} ]_o^\pi \quad $$$$ \Rightarrow A = ( - \cos \pi ) - ( - \cos 0)$$$$ \Rightarrow A = (1) - ( - 1)$$$$ \Rightarrow A = 2$$ Therefore, the area bounded by the curve $y = \operatorname{sinx} $, $x$-axis, $y$-axis and the line $x = \pi $ is $2$ sq.units

Rough sketch of the Curve:

Before finding the area of any curve, we get the rough sketch of curve so that it might make find the area easily. For example if we know the symmetric parts of the curve we can find the area of one symmetric part and multiply it with total number of symmetrical parts to get total area.