Page:Calculus Made Easy.pdf/234

${\begin{aligned}&=b{\bigl [}\log _{\epsilon }(x+a){\bigr ]}_{0}^{x_{1}}+C\\&=b{\bigl [}\log _{\epsilon }(x_{1}+a)-\log _{\epsilon }(0+a){\bigr ]}\\&=b\log _{\epsilon }{\frac {x_{1}+a}{a}}.\quad Ans.\end{aligned}}$

N.B–Notice that in dealing with definite integrals the constant $C$ always disappears by subtraction.

Let it be noted that this process of subtracting one part from a larger to find the difference is really a common practice. How do you find the area of a

Fig. 56.

plane ring (Fig. 56), the outer radius of which is $r_{2}$ and the inner radius is $r_{1}$? You know from mensuration that the area of the outer circle is $\pi r_{2}^{2}$; then you find the area of the inner circle, $\pi r_{1}^{2}$; then you subtract the latter from the former, and find area of ring $=\pi (r_{2}^{2}-r_{1}^{2})$; which may be written

$\pi (r_{2}+r_{1})(r_{2}-r_{1})$

= mean circumference of ring × width of ring.