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### Section 5.6 : Definition of the Definite Integral

4. Determine the value of $$\displaystyle \int_{{11}}^{6}{{9f\left( x \right)\,dx}}$$ given that $$\displaystyle \int_{6}^{{11}}{{f\left( x \right)\,dx}} = - 7$$.

Show Solution
There really isnâ€™t much to this problem other than use the properties from the notes of this section until we get the given interval at which point we use the given value.

\begin{align*}\int_{{11}}^{6}{{9f\left( x \right)\,dx}} & = 9\int_{{11}}^{6}{{f\left( x \right)\,dx}} & \hspace{0.5in}& {\mbox{Property 3}}\\ & = - 9\int_{6}^{{11}}{{f\left( x \right)\,dx}} & \hspace{0.5in} & {\mbox{Property 1}}\\ & = - 9\left( { - 7} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{63}} & & \end{align*}