Area of the Region Bounded by a Curve, x-axis and the Lines x=a, x=b

Case (i): curve above the $x$-axis. Let $y=f(x)$, $a\le x\le b$, be a continuous curve lying entirely above the $x$-axis (in the first or second quadrant) between $x=a$ and $x=b$, so $y\ge0$ throughout. Viewing the region bounded by the curve, the $x$-axis, and the ordinates $x=a,\,x=b$ in the positive $y$-direction, divide it into thin vertical strips of height $y$ and width $\Delta x$. The area $A$ is the limit of the sum of the strip areas:

Case (ii): curve below the $x$-axis. If $y=f(x)$ lies entirely below the $x$-axis (third/fourth quadrant) on $[a,b]$, so $y\le0$ throughout, each strip has height $-y$ (a positive length), giving

Case (iii): curve crosses the $x$-axis. If $y=f(x)$ takes both signs on $[a,b]$, divide $[a,b]$ at the crossing points $c_1,c_2,\ldots,c_k$ into subintervals where $f$ keeps a constant sign, apply Cases (i)/(ii) on each piece, and add the (positive) areas:

$A=\left|\int_a^{c_1}f(x)\,dx\right|+\left|\int_{c_1}^{c_2}f(x)\,dx\right|+\cdots+\left|\int_{c_k}^{b}f(x)\,dx\right|.$ …