3.4a Definite Integral of f(x) from x=a to x=b

3.4a Definite Integral of f(x) from x=a to x=b

Example:
Evaluate each of the following.
$\begin{array}{l}\text{(a)}{\int }_{-1}^{0}\left(3{x}^{2}-2x+5\right)dx\\ \text{(b)}{\int }_{0}^{2}{\left(2x+1\right)}^{3}dx\end{array}$

Solution:
$\begin{array}{l}\text{(a)}{\int }_{-1}^{0}\left(3{x}^{2}-2x+5\right)dx\\ ={\left[\frac{3{x}^{3}}{3}-\frac{2{x}^{2}}{2}+5x\right]}_{-1}^{0}\\ ={\left[{x}^{3}-{x}^{2}+5x\right]}_{-1}^{0}\\ =0-\left[{\left(-1\right)}^{3}-{\left(-1\right)}^{2}+5\left(-1\right)\right]\\ =0-\left(-1-1-5\right)\\ =7\\ \\ \\ \text{(b)}{\int }_{0}^{2}{\left(2x+1\right)}^{3}dx\\ ={\left[\frac{{\left(2x+1\right)}^{4}}{4\left(2\right)}\right]}_{0}^{2}\\ ={\left[\frac{{\left(2x+1\right)}^{4}}{8}\right]}_{0}^{2}\\ =\left[\frac{{\left(2\left(2\right)+1\right)}^{4}}{8}\right]-\left[\frac{{\left(2\left(0\right)+1\right)}^{4}}{8}\right]\\ =\frac{625}{8}-\frac{1}{8}\\ =78\end{array}$