Area Under a Curve
Calculus & Advanced MathThe 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.
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.
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.
Definition
The area under $y = f(x)$ from $a$ to $b$ (for $f \ge 0$) equals $\int_a^b f(x)\, dx$. If $f$ dips below the $x$-axis, the integral counts that portion as negative (net signed area). For total area, integrate $|f(x)|$.
Example
Area between $y = \sin(x)$ and $x$-axis from $0$ to $2\pi$: $\int_0^\pi \sin(x)\, dx - \int_\pi^{2\pi} \sin(x)\, dx = 2 + 2 = 4$ (using absolute value of each piece).
Key Insight
The distinction between "net signed area" (definite integral) and "total area" (integral of |f|) is important in applications like displacement vs. distance.
Definition
The area between two curves $f$ and $g$ on $[a,b]$ is $\int_a^b |f(x) - g(x)|\, dx$. More generally, area of a region $D$ 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
Green's theorem gives a planimeter algorithm: trace the boundary of any closed region and the integral automatically returns the enclosed area. This is used in CAD and GIS software.
Key Insight
Area computation is the gateway to higher-dimensional volume, flux, and probability calculations, all unified by the general theory of measure and integration.