Section 5.6 : Definition of the Definite Integral
7. Determine the value of \( \displaystyle \int_{{ - 4}}^{{20}}{{f\left( x \right)\,dx}}\) given that \( \displaystyle \int_{{ - 4}}^{0}{{f\left( x \right)\,dx}} = - 2\), \( \displaystyle \int_{{31}}^{0}{{f\left( x \right)\,dx}} = 19\) and \( \displaystyle \int_{{20}}^{{31}}{{f\left( x \right)\,dx}} = - 21\).
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Start SolutionFirst we need to use Property 5 from the notes of this section to break up the integral into three integrals that use the same limits as the integrals given in the problem statement.
Note that we won’t worry about whether the limits are in correct place at this point.
\[\int_{{ - 4}}^{{20}}{{f\left( x \right)\,dx}} = \int_{{ - 4}}^{0}{{f\left( x \right)\,dx}} + \int_{0}^{{31}}{{f\left( x \right)\,dx}} + \int_{{31}}^{{20}}{{f\left( x \right)\,dx}}\] Show Step 2Finally, all we need to do is use Property 1 from the notes of this section to interchange the limits on the second and third integrals so they match up with the limits on the given integral. We can then use the given values to determine the value of the integral.
\[\int_{{ - 4}}^{{20}}{{f\left( x \right)\,dx}} = \int_{{ - 4}}^{0}{{f\left( x \right)\,dx}} - \int_{{31}}^{0}{{f\left( x \right)\,dx}} - \int_{{20}}^{{31}}{{f\left( x \right)\,dx}} = - 2 - \left( {19} \right) - \left( { - 21} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{0}\]