Jamal wrote the following Riemann sum to estimate the area under $f (x) = 3x^2 - 2$.  

$\displaystyle \sum _ { i = 0 } ^ { 9 } \frac { 1 } { 2 } f ( - 3 + \frac { 1 } { 2 } i )$

1. Draw a sketch of the region. How many rectangles did he use?

   Hint (a):

   General form of left-endpoint Riemann Sum:

   $\displaystyle \sum_{i=0}^{n-1}\Delta xf(a+\Delta xi)$

   Answer (a):

   $n − 1 = 9 n = 10$ There are $10$ rectangles.

   Step 1(a):

   The function is $f(x) = 3x^2 − 2$.

   Step 2(a):

   $a = −3$ This is where the region starts.

   Step 3(a):

   $\Delta x=\frac{1}{2}$

   $\text{Each rectangle has a width of }\frac{1}{2}.$

2. For what domain of $f$ did Jamal estimate the area?

   Hint (b):

   Refer to the graph in part (a).

3. Use the summation feature of your calculator to approximate the area using Jamal’s Riemann sum.