The area under the function $f(x)=3x^3+1$ on the interval from $1$ to $3$ is to be approximated using $10$ left-endpoint rectangles. 

1. Write the summation notation that will represent the area.

   Step 1(a):

   Sketch the function.

   Step 2(a):

   Draw $10$ rectangles between $x = 1$ and $3$.

   Step 3(a)

   What is the width for each rectangle? What are the $x$-values for the heights?

   Answer 1(a):

   $\frac{3-1}{10}=\frac{2}{10}=0.2$

   $x_{0} = 1$; $x_{1} = 1.2$; $x_{2} = 1.4$; $x_{3} = 1.6$; $\dots$; $x_{9} = 2.8$

   Step 4(a):

   What expression needs to go into $f(x)$ to convert the integer values to $1$, $1.2$, $1.4$, $1.6$, $\dots$, $2.8$?

   Answer 2(a):

   $0.2k+1$

   Step 5(a):

   Write the summation now. Be careful with your $k$-values.

2. Make a small change to your answer in part (a) so that the sum will find the area using right-endpoint rectangles.

   Hint (b):

   What $k$-values should you start and stop with?

3. Make a small change to your answer in part (a) so that the sum will find the area using midpoint rectangles.

   Hint (c):

   Change the inside function to: $0.2k + 1.1$ so that it is starting at the middle of each rectangle.