​Consider the area under the curve of $f\left(x\right)=\sqrt{x^2+2}$ on the interval $3\le x\le5$. 

1. Write out the sums you would need to determine the right endpoint and left endpoint rectangle approximations for the area under the curve using four rectangles. Do not evaluate the sums. Leave the expressions in expanded form.

2. Express each of the sums using sigma notation. You should be able to express the different sums by just changing the indices of each summation.

   Hint (b):

   The sequence for the '$x_k$'s is $3$, $3.5$, $4$, $4.5$, 5.
   Write a linear expression for this sequence.
   Then, for the sigma notation, write an expression for $f(x_k)$.

3. Explain how the indices relate to right endpoint and left endpoint rectangles.