Areas delimited by two functions

If now we want to determine the area delimited by two functions:

we will decompose the problem into two parts: first we will calculate the area delimited by the top graph and the $x$ axis and will subtract the area delimited by the lower graph.

So, if the top graph is $(x,f(x))$ and the lower one is $(x,g(x))$: $$ A=\int_a^b f(x) \ dx - \int_a^b g(x) \ dx$$

We are going to calculate the area delimited by the functions $f(x)=\sin\Big( \dfrac{\pi}{2} \Big)$ and $g(x)=x$ in the interval $[0,1]$. Let's draw the functions to know which is the top graph and which the bottom one:

We see, then, that the top graph is $f(x)$ and the low one is $g(x)$. Therefore the area between two functions $[-1,1]$ is:

$$\begin{array}{rl} A =&\int_0^1 f(x)\ dx-\int_0^1 g(x) \ dx = \int_0^1 \sin\Big(\dfrac{\pi}{2}x\Big) \ dx- \int_0^1 x \ dx \\ =& \dfrac{2}{\pi}\Big[-\cos\Big(\dfrac{\pi}{2}x \Big)\Big]_0^1-\Big[\dfrac{x^2}{2}\Big]_0^1= \dfrac{2}{\pi}(0+1)-\dfrac{1}{2}=\dfrac{4-\pi}{2\pi}u^2 \end{array}$$