  ### Home > PC3 > Chapter 4 > Lesson 4.1.1 > Problem4-10

4-10.

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

Carefully sketch the function for the given interval. Part (a) below asks for four rectangles, so divide the interval into four sections. Compute the function values for the endpoints of the sections. Use these values to make an accurate sketch.

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.

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.