Question

Evaluate the following integrals as limit of sum 1) integral 0 to 2 (2x+1) dx
​2) integral 3 to 5 (2-x) dx

Answer 1

$$\begin{gathered} \left(1\right). \\ \mathrm{Let}\mathrm{I}={\int }_0^2\left(2x+1\right)dx \\ ={\left[x^2+x\right]}_0^2 \\ =\left[\left(4+2\right)-0\right] \\ =6 \end{gathered}$$

$$\begin{gathered} \left(2\right). \\ \mathrm{Let}\mathrm{I}={\int }_3^5\left(2-x\right)dx \\ ={\left[2x-\frac{x^2}{2}\right]}_3^5 \\ =\left[\left(10-\frac{25}{2}\right)-\left(6-\frac{9}{2}\right)\right] \\ =10-\frac{25}{2}-6+\frac{9}{2} \\ =4-8 \\ =-4 \end{gathered}$$