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a mathematical instrument for measuring the areas of plane figures of arbitrary shape. The most widespread type is the polar planimeter, which was invented by the Swiss mathematician J. Amsler in 1854.
The principle of operation of such a planimeter is shown in Figure 1. When the oriented line segment AB moves in the plane, the area of the figure swept out by it is equal to the product of the length of AB and the length of the arc described by the segment’s midpoint. The quantity σ is equal to the length of the arc of rotation of the integrating wheel if the wheel is placed at the midpoint of AB and its axis coincides with the segment. The polar planimeter (Figure 2) has a polar arm and a tracing arm, which are pivoted at the point B. The polar arm can be rotated about the pole pivot O. The integrating wheel and the counting mechanism are located on a special carriage that can slide along the tracing arm, thereby changing the length of the arm.
In every planimeter the function of the sweeping line segment is performed by the tracing arm, at one end of which is fastened a pointer for tracing out the figure’s outline and the other end of which moves along a “directrix.” The directrix is a straight line in a linear planimeter, a circle in a polar planimeter, and a point in a radial planimeter. Linear and polar planimeters are used to compute areas bounded by curves given graphically and to calculate integrals of the form
Radial planimeters permit integrals of the form
to be found if the curve r = f (ϕ) is drawn in polar coordinates.