Page:Calculus Made Easy.pdf/233

 Now reckon out the area beneath the curve by counting the little squares below the line, from $$x=0$$ as far as $$x=12$$ on the right. There are $$18$$ whole squares and four triangles, each of which has an area equal to $$1\tfrac{1}{2}$$ squares; or, in total, $$24$$ squares. Hence $$24$$ is the numerical value of the integral of $$\dfrac{x}{3}\, dx$$ between the lower limit of $$x=0$$ and the higher limit of $$x=12$$.

As a further exercise, show that the value of the same integral between the limits of $$x=3$$ and $$x=15$$ is $$36$$.

(2) Find the area, between limits $$x=x_1$$ and $$x=0$$, of the curve $$y = \dfrac{b}{x + a}$$.