Page:Calculus Made Easy.pdf/230

 and this will be the whole area from $$0$$ up to any value of $$x$$ that we may assign.

Therefore, the larger area up to the superior limit $$x_2$$ will be

and the smaller area up to the inferior limit $$x_1$$ will be

Now, subtract the smaller from the larger, and we get for the area $$S$$ the value,

This is the answer we wanted. Let us give some numerical values. Suppose $$b = 10$$, $$a = 0.06$$, and $$x_2 = 6$$ and $$x_1 = 6$$. Then the area $$S$$ is equal to

Let us here put down a symbolic way of stating what we have ascertained about limits:

where $$y_2$$ is the integrated value of $$y\, dx$$ corresponding to $$x_2$$, and $$y_1$$ that corresponding to $$x_1$$.