3.4b Laws of Definite Integrals

3.4b Laws of Definite Integrals

Example:
Given that ${\int }_{3}^{7}f\left(x\right)dx=5$ , find the values for each of the following:

$\begin{array}{l}\text{(a)}{\int }_{3}^{7}6f\left(x\right)dx\\ \text{(b)}{\int }_{3}^{7}\left[3-f\left(x\right)\right]dx\\ \text{(c)}{\int }_{7}^{3}2f\left(x\right)dx\\ \text{(d)}{\int }_{3}^{4}f\left(x\right)dx+{\int }_{4}^{5}f\left(x\right)dx+{\int }_{3}^{7}f\left(x\right)dx\\ \text{(e)}{\int }_{3}^{7}\frac{f\left(x\right)+7}{2}dx\end{array}$

Solution:
$\begin{array}{l}\text{(a)}{\int }_{3}^{7}6f\left(x\right)dx=6{\int }_{3}^{7}f\left(x\right)dx\\ \text{}=6\left(5\right)=30\\ \\ \text{(b)}{\int }_{3}^{7}\left[3-f\left(x\right)\right]dx={\int }_{3}^{7}3dx-{\int }_{3}^{7}f\left(x\right)dx\\ ={\left[3x\right]}_{3}^{7}-5\\ =\left[3\left(7\right)-3\left(3\right)\right]-5\\ =7\\ \\ \text{(c)}{\int }_{7}^{3}2f\left(x\right)dx=-{\int }_{3}^{7}2f\left(x\right)dx\\ =-2{\int }_{3}^{7}f\left(x\right)dx\\ =-2\left(5\right)\\ =-10\\ \\ \text{(d)}{\int }_{3}^{4}f\left(x\right)dx+{\int }_{4}^{5}f\left(x\right)dx+{\int }_{3}^{7}f\left(x\right)dx\\ ={\int }_{3}^{7}f\left(x\right)dx\\ =5\\ \\ \text{(e)}{\int }_{3}^{7}\frac{f\left(x\right)+7}{2}dx={\int }_{3}^{7}\left[\frac{1}{2}f\left(x\right)+\frac{7}{2}\right]dx\\ ={\int }_{3}^{7}\frac{1}{2}f\left(x\right)dx+{\int }_{3}^{7}\frac{7}{2}dx\\ =\frac{1}{2}{\int }_{3}^{7}f\left(x\right)dx+{\left[\frac{7x}{2}\right]}_{3}^{7}\\ =\frac{1}{2}\left(5\right)+\left[\frac{7\left(7\right)}{2}-\frac{7\left(3\right)}{2}\right]\\ =\frac{5}{2}+14\\ =16\frac{1}{2}\end{array}$

1 thought on “3.4b Laws of Definite Integrals”

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