Calculus I - Definition of the Definite Integral

Show Mobile Notice Show All Notes Hide All Notes

Mobile Notice

You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

Section 5.6 : Definition of the Definite Integral

Back to Problem List

6. Determine the value of \( \displaystyle \int_{2}^{9}{{f\left( x \right)\,dx}}\) given that \( \displaystyle \int_{5}^{2}{{f\left( x \right)\,dx}} = 3\) and \( \displaystyle \int_{5}^{9}{{f\left( x \right)\,dx}} = 8\).

Show All Steps Hide All Steps

Start Solution

First we need to use Property 5 from the notes of this section to break up the integral into two 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.

\[\int_{2}^{9}{{f\left( x \right)\,dx}} = \int_{2}^{5}{{f\left( x \right)\,dx}} + \int_{5}^{9}{{f\left( x \right)\,dx}}\] Show Step 2

Finally, all we need to do is use Property 1 from the notes of this section to interchange the limits on the first integral 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.

\[\int_{2}^{9}{{f\left( x \right)\,dx}} = - \int_{5}^{2}{{f\left( x \right)\,dx}} + \int_{5}^{9}{{f\left( x \right)\,dx}} = - \left( 3 \right) + 8 = \require{bbox} \bbox[2pt,border:1px solid black]{5}\]