Area of a Curvilinear Trapezoid
What is a Curvilinear Trapezoid?
A curvilinear trapezoid is a plane figure bounded by:
- The graph of a continuous non-negative function $y = f(x)$ (the curved top)
- The $x$-axis (the straight bottom)
- Two vertical lines $x = a$ and $x = b$ (the lateral sides)
It is called "curvilinear" because at least one of its sides is a curve rather than a straight line.
Formal Definition of the Area
Let $f(x)$ be a continuous and non-negative function on the interval $[a, b]$. The area of the curvilinear trapezoid bounded by $y = f(x)$, $y = 0$, $x = a$, and $x = b$ is given by the definite integral:
$$S = \int_{a}^{b} f(x)\, dx$$
This is one of the most fundamental applications of the definite integral and serves as the geometric motivation for its definition.