Changing the Limits of Integration

Go back to  'Definite Integration'

(12) Sometimes, it is convenient to change the limits of integration into some other limits. For example, suppose we have to add two definite integrals \({I_1}{\rm{ and }}{I_2}\) ; the limits of integration for these integrals are different. If we could somehow change the limits of \({I_2}\) into those of \({I_1}\) or vice-versa, or in fact change the limits of both \({I_1}{\rm{ and }}{I_2}\) into a third (common) set of limits, the addition could be accomplished easily.

Suppose that \(I = \int\limits_a^b {f(x)\;dx} \) . We need to change the limits (a to b) to (a' to b'). As x varies from a to b, we need a new variable t (in terms of x) which varies from a' to b'.

As described in the figure above, the new variable t is given by,

\[t = a' + \left( {\frac{{b' - a'}}{{b - a}}} \right)(x - a)\]

Thus,

\[\begin{align}&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,dt = \frac{{b' - a'}}{{b - a}}dx\\ &\;\Rightarrow \quad I = \int\limits_a^b {f(x)\,dx} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, &\qquad\quad= \int\limits_{a'}^{b'} {f\left( {a + \left( {\frac{{b - a}}{{b' - a'}}} \right)(t - a')} \right)\left( {\frac{{b - a}}{{b' - a'}}} \right)\,dt} \end{align}\]

The modified integral has the limits \((a'\,\,{\rm{to }}\;b')\) . A particular case of this property is modifying the arbitrary integration limits (a to b) to (0 to 1) i.e., \(a' = 0\,\,{\rm{and }}\;b' = 1\) . For this case,

\[\begin{array}{l}I = \int\limits_a^b {f(x)\;dx} \\\,\,\,\, = (b - a)\int\limits_0^1 {f(a + (b - a)t)\,dt} \end{array}\]