12 



MATHEMATICS 



page 10, applied. For the method of figuring areas of polygons 

 by double longitudes, see page 60. 



AREA INCLUDED BETWEEN A STRAIGHT LINE AND 



A CURVE 



Method by Selected Ordinates. Draw perpendiculars on 

 A B from the points of the curve at which its direction changes 



appreciably, and consider the portion of the curve between 

 two consecutive perpendiculars to be a straight line. The 

 figure is then treated as if divided into a number of trape- 

 zoids, whose areas can be computed by the rules already given. 

 Trapezoidal Rule. The ordinates are measured at regular 

 intervals d along the straight line as shown. The area is then 

 equal to 



JO 



A = 



in which 2A is the sum of all the intermediate ordinates. 



EXAMPLE. If the ordinates from the straight line AB to 

 the curved boundary DC, are 19, 18, 14, 12, 13, 17, and 23 li., 

 respectively, and are at equal distances of 50 li., what is the 

 area included between the curved boundary and the straight 

 line? 



/1Q _L_ OO \ 



SOLUTION. Area ABCD= ( !1 + 18+14 + 12+13+17J 

 X 50 = 4,750 sq. li. 



