In
calculus, the
definite integral of a
function f(x) can be
numerically approximated as a discrete
sum by
partitioning the interval of integration into small uniform intervals and approximating the function's value on each interval as the average of the values at its endpoints: \int_{a}^{b} f(x)\, dx \approx \sum_{k=1}^{N} \tfrac12 \bigl( f(x_{k-1}) + f(x_{k}) \bigr) \Delta x, where N is the number of intervals, x_0 = a, x_N = b, and \Delta x = (b - a) / N. Graphically, this amounts to approximating the region under the
graph of the function by a collection of trapezoids, so this method is called the
trapezoidal rule. When any rectangle is viewed in
perspective from a position which is centered on one axis but not the other, it appears to be an isosceles trapezoid, called the
keystone effect because arch
keystones are commonly trapezoidal. For example, when a rectangular building
façade is photographed from the ground at a position directly in front using a
rectilinear lens, the image of the building is an isosceles trapezoid. Such photographs sometimes have a "keystone transformation" applied to them to recover rectangular shapes.
Video projectors sometimes apply such a keystone transformation to the recorded image before projection, so that the image projected on a flat screen appears undistorted. Trapezoidal doors and windows were the standard style for the
Inca, although it can be found used by earlier cultures of the same region and did not necessarily originate with them. An almena, a
battlement feature characteristic of
Moorish architecture, is trapezoidal. Michaelangelo's redesign of the
Piazza del Campidoglio (see photograph at right) incorporated a trapezoid surrounding an ellipse, giving the effect of a square surrounding a circle when seen foreshortened at ground level.
Cinematography takes advantage of trapezoids in the opposite way, to produce an excessive foreshortening effect from the camera viewpoint, giving the illusion of greater depth to a room in a movie studio than the set physically has. Trapezoids were also used to produce the visual distortions of
Caligarism.
Canals and drainage
ditches commonly have a trapezoidal cross-section. In biology, especially
morphology and
taxonomy, terms such as
trapezoidal or
trapeziform commonly are useful in descriptions of particular organs or forms. Trapezoids are sometimes used as a graphical symbol. In
circuit diagrams, a trapezoid is the symbol for a
multiplexer. An isosceles trapezoid is used for the shape of road signs, for example, on secondary highways in
Ontario, Canada. == Non-Euclidean geometry ==