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

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

   Step 1(a):

   Accurately sketch the function for $x=1$ to $x=3$.

   Step 2(a):

   Draw $10$ rectangles between $x=1$ and $3$. What is the width of each rectangle?

   Step 3(a):

   What are the $x$-values for the endpoints of the rectangles?

   Step 4(a):

   Write a linear expression for the endpoints: $1,1.2,1.4,1.6,...$

   Step 5(a):

   Substitute the linear expression you wrote in step 4 into the function for $x$. This will be the argument of your summation notation.

2. Modify your answer in part (a) to calculate the area using right endpoint rectangles.

   Hint (b):

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