MEASUREMENTS 



291 



(P) 



(S) 



Fio. 197. F 



Area of a parabola. Area of a sinusoidal curve. 



Area of the sinusoidal curve (fig. 198) : 



S =2. 



218. The problem of the determination of the areas of plane 

 surfaces, that is to say, of the curves traced in a plane and whose 

 equation usually is : 



y=f (*). 



is called the problem of the quadratures. It resolves itself always 

 into the integration of the equation y =f (x). Mechanical in- 

 tegrators or planimeters have been constructed, by which areas, 

 or, in other words work done can be measured. They comprise 

 systems of jointed levers, turning round a fixed axis, and carrying 

 a graduated cylinder and a tracing point. Whilst the tracer 

 passes over the boundary of the area to be measured, the cylinder 

 turns through an angle proportional to the area and gives its 

 value. Amsler's planimeter is one of the best known. The pre- 

 sent model consists of two rods AP and AT (fig. 199), jointed at 



