Area Under a Curve
The area under a curve is the region between the graph of a function and the x-axis over an interval, calculated using a definite integral.
Formula
A = \int_a^b f(x)\, dx
Definition
The area under a curve is the amount of space enclosed between the curve and the x-axis over a given interval, like measuring the region under a hill on a graph. For $f \ge 0$, the area under $y = f(x)$ from $a$ to $b$ equals $A = \int_a^b f(x)\, dx$; if $f$ dips below the $x$-axis, the integral counts that portion as negative (net signed area), so for total area you integrate $|f(x)|$ instead. More generally, the area between two curves $f$ and $g$ on $[a,b]$ is $\int_a^b |f(x) - g(x)|\, dx$, and area of a region in $\mathbb{R}^2$ is given by the double integral $\iint_D 1\, dA$; Green's theorem connects boundary integrals to area via $A = \frac{1}{2} \oint (x\, dy - y\, dx)$.
Example
The area under $y = 4$ from $x = 0$ to $x = 5$ is a rectangle, $4 \times 5 = 20$ square units; for a curved graph you use integration instead. The area between $y = \sin(x)$ and the $x$-axis from $0$ to $2\pi$ needs each piece's absolute value: $2 + 2 = 4$. Green's theorem gives a planimeter algorithm: trace the boundary of any closed region and the integral automatically returns the enclosed area, used in CAD and GIS software.
Key Insight
Area under a curve is how integration connects to real-world measurements: total distance traveled, total accumulated rainfall, total work done by a force. The distinction between "net signed area" (definite integral) and "total area" (integral of $|f|$) is important in applications like displacement versus distance, and area computation is the gateway to higher-dimensional volume, flux, and probability calculations, all unified by the general theory of measure and integration.