The area A of the parabolic segment enclosed by the parabola and the chord is therefore =. This formula can be compared with the area of a triangle: ⁠ 1 / 2 ⁠ bh. In general, the enclosed area can be calculated as follows. First, locate the point on the parabola where its slope equals that of the chord.

The general form of a parabola is written as \[ax^2+bx+cy+d=0 \label{para1} \] or \[ay^2+bx+cy+d=0.\label{para2} \] Equation \ref{para1} represents a parabola that opens either up or down. Equation \ref{para2} represents a parabola that opens either to the left or to the right.

We can use this idea to define a double integral over a general bounded region. To evaluate an iterated integral of a function over a general nonrectangular region, we sketch the region and express it as a Type I or as a Type II region or as a union of several Type I or Type II regions that overlap only on their boundaries.

We start by finding the area between two curves that are functions of \(\displaystyle x\), beginning with the simple case in which one function value is always greater than the other. We then look at cases when the graphs of the functions cross. Last, we consider how to calculate the area between two curves that are functions of \(\displaystyle ...

A mirror that has a cross-sectional parabolic shape has the property that all light shone directly towards it will be reflected towards the focus of the parabola. A parabolic reflector is a mirror or reflective device that uses this property to concentrate reflected light onto a single point.