Question

If \(\int_1^5 f (x)dx = 12\) and \(\int_4^5 f (x)dx = 3.6\), find \(\int_1^4 f (x)dx\).

Short answer

The value of \(\int_1^4 f (x)dx\) is \(8.4.\)

Step-by-step solution

Property Used

\(f\) to be an integrable function. Then, \(\int_a^c f (x)dx + \int_c^b f (x)dx = \int_a^b f (x)dx\) where, the limits \(a,b\)and \(c\) are constants and \(a < c < b\).

Calculation of the integral

The integrals are\(\int_1^5 f (x)dx = 12\)and\(\int_4^5 f (x)dx = 3.6\)

Obtain the value of the integral,\(\int_1^4 f (x)dx\)as follows.

Consider the limit values as\(a = 1,b = 5\)and\(c = 4\)so that,\(a < c < b\).

Hence, as per the mentioned property,

\(\int_1^4 f (x)dx + \int_4^5 f (x)dx = \int_1^5 f (x)dx\)

\(\int_1^4 f (x)dx = \int_1^5 f (x)dx - \int_4^5 f (x)dx{\kern 1pt} {\kern 1pt} ........(1)\)

Substitute the values in the integral

Substitute the values of\(\int_1^5 f (x)dx = 12\)and\(\int_4^5 f (x)dx = 3.6\)in the above obtains equation (1) and compute the value of\(\int_1^4 f (x)dx\)

\(\begin{aligned}{l}\int_1^4 f (x)dx &= \int_1^5 f (x)dx - \int_4^5 f (x)dx\\ &= 12 - 3.6\\ &= 8.4\end{aligned}\)

So, the value of \(\int_1^4 f (x)dx\) is \(8.4\)