Area Bounded by Curve and Axes

In the previous concept, we learned that if the function $f(x) \geq 0 \forall x \in[a, b]$ then $\int_a^b f(x) d x$ represents the area bounded by $y=f(x), x$-axis and lines $x=a$ and $x=b$.

If the function $f(x) \leq 0 \forall x \in[a, b]$, then the area by bounded $y=f(x), x$-axis and lines $x=a$ and $x=b$ is $\left|\int_a^b f(x) d x\right|$.

Area along Y-axis

The area by bounded $x=g(y)[$ with $g(y)>0], y$-axis and the lines $y=a$ and $y=b$ is $\int_a^b x d y=\int_a^b g(y) d y$

Area of Piecewise Function

If the graph of the function f(x) is of the following form, then.

then $\int_a^b f(x) d x$ will equal $A_1-A_2+A_3-A_4$ and not $A_1+A_2+A_3+A_4$.
If we need to evaluate $A_1+A_2+A_3+A_4$ (the magnitude of the bounded area), we will have to calculate
$
\underbrace{\int_a^x f(x) d x}_{\mathrm{A}_1}+\underbrace{\left|\int_x^y f(x) d x\right|}_{\mathrm{A}_2}+\underbrace{\int_y^z f(x) d x}_{\mathrm{A}_3}+\underbrace{\left|\int_z^b f(x) d x\right|}_{\mathrm{A}_4}
$

The area bounded by the curve when the curve intersects the X-axis

The graph $y=f(x) \forall x \in[a, b]$ intersects the $x$-axis at $x=c$.
If the function $f(x) \geq 0 \forall x \in[a, c]$ and $f(x) \leq 0 \forall x \in[c, b]$ then the area bounded by curve and $x$-axis, between lines $x=a$ and $\mathrm{x}=\mathrm{b}$ is

$\int_a^b|f(x)| d x=\int_a^c f(x) d x-\int_c^b f(x) d x$

The area is bounded by the curves $y=f(x), y=g(x)$, and the lines $x=a$ and $x=b$, and it is given that $f(x) \leq g(x)$.

From the figure, it is clear that, 

Area of the shaded region = Area of the region ABEF -  Area of the region ABCD

$\int_a^b g(x) d x-\int_a^b f(x) d x=\int_a^b(\underbrace{g(x)}_{\begin{array}{c}\text { upper } \\ \text { curve }\end{array}}-\underbrace{f(x)}_{\begin{array}{c}\text { lower } \\ \text { curve }\end{array}}) d x$

Area bounded by the curves $y=f(x), y=g(x)$ which intersect each other in the interval [a, b]
First find the point of intersection of these curves $y=f(x)$ and $y=g(x)$ by solving the equation $f(x)$ $=g(x)$, let the point of intersection be $x=c$

Area of the shaded region  

$=\int_a^c\{f(x)-g(x)\} d x+\int_c^b\{g(x)-f(x)\} d x$

When two curves intersect more than one point

Area bounded by the curves $y=f(x), y=g(x)$ which intersect each other at three points at $x=a, x=$ $b$ and $x=c$.

To find the point of intersection, solve $f(x)=g(x)$.
For $x \in(a, c), f(x)>g(x)$ and for $x \in(c, b), g(x)>f(x)$.

Area bounded by curves,

$\begin{aligned} \mathrm{A} & =\int_a^b|f(x)-g(x)| d x \\ & =\int_a^c(f(x)-g(x)) d x+\int_c^b(g(x)-f(x)) d x\end{aligned}$