108 SUPPLEMENT TO CHAP. VH. [OHAP. H. 



aide y in passing from one position to the consecutive or very next. We 



have, then, d a = b dx , and if we integrate this expression, that is, 



sum up all the d a's, we have the area of the triangle. Therefore, 



= / bdx 



b x d x b a? 



= b X TTT 



h 



where O is the constant of integration, which we must never forget to annex. 

 Now, in the present case we wish to sum up all the areas d a, or " integrate," 

 between the limits x o and * = h. But for x = o, A must be zero, and 

 hence we have O = o for the condition that x starts from the base. If in 

 addition to this condition we make * = A, we have the sum of all the areas 

 between x = o and x = A. 



, T b h b h , , , 

 A = 6 h -- = , as should be. 

 2 2 



The above reasoning is somewhat prolix. 



If we thoroughly appreciate that d x is the difference between two con- 

 teoulive values of *, we see at once that we obtain the limiting value of the 

 rectangle directly by multiplying its base by d x. The sum of all these 

 must be the area. This conception of d x enables us to curtail much of our 

 reasoning. 



Let us take the same problem again, but this time take the axis through 

 the centre of gravity of the triangle ; that is, at %h above the base. Then 

 for the base y at any distance x above this axis, we have 



2 2, Ix 



-h-x-.y::h:b, ory = -b-. 



Multiply this by d x upon the above conception of d x, and we have at 

 once not for the rectangle upon y, but for its limiting value, that is, for the 

 area of that portion of the rectangle included within the triangle, 



2. , b x dx 

 d a = y dx = -bdx -- j . 

 o h 



Integrating this, then, we have 



where O is a constant to be determined by the limits as before. For one 



limit, x = h, and hence we have 

 3 



*' n**** 



2 



For the other limit, z = + A, and hence we have 

 8 



If we subtract the first from the second, O disappears, and we have A = 



A'- A'= ?-b h = ^b h, as before. 



2 



We might also have integrated first between the limits x = and * = A. 



3 



