Use a Riemann sum with $20$ rectangles to approximate the following integrals. Then use the numerical integration feature of your graphing calculator to check your answers.

1. $\int _ { 0 } ^ { 4 } ( 2 - 4 x ^ { 3 / 2 } ) d x$

   Hint (a):

   $f(x)\text{ is the integrand: }f(x)=2-4x^{\frac{3}{2}}$

   $\text{The area is being evaluated on }0\leq x\leq 4...\text{ so }\Delta x= \frac{4}{20}=\frac{1}{5}.$

   Answer (a)

   $\text{Riemann approximation: }\approx \sum_{i\rightarrow 0}^{19}\frac{1}{5}f\left ( 0+\frac{i}{5} \right )=\sum_{i\rightarrow 0}^{19}\frac{1}{5}\left ( 2-4\left ( \frac{i}{5} \right )^{\frac{3}{2}} \right )=-40.04$

   $\text{Your calculator should reveal that }\int_{0}^{4}\left ( 2-4x^{\frac{3}{2}} \right )=-43.2.$

2. $\int _ { 1 } ^ { 8 } \sqrt { 4 x + 3 } d x$

   Hint (b):

   Refer to part (a).