Suppose we want to find the area under the curve $f\left(x\right)$ over the interval $1 ≤ x ≤ 10$ using $25$ sub-intervals. What is the width of each rectangle?

1. What will the height of the first rectangle be if we use left-endpoint rectangles?

   Hint (a):

   $x_0=1→\text{height of rectangle}=f\left(1\right)$

2. What will be the height of the first rectangle if we use right-endpoint rectangles?

   Hint (b):

   Height of $x_{1}$.

3. What will be the height of the last rectangle if we use right-endpoint rectangles?

   Hint (c):

   For right-endpoint, we start at $x_{1}$ and go to the end. So you want the height at that $x_{\text{end}}$.

4. What will be the height of the last rectangle if we use left-endpoint rectangles?

   Hint (d):

   For left-endpoint, we start at $x_{0}$ and go to the $x_{\text{end}-1}$. So you want the $\text{height}-f\left(\text{end}-\text{width}\right)$.

5. How would your answers change if the interval were $B ≤ x ≤ E$ and the width of each rectangle were $W$?

   Answer (e):

   $x_{\text{One Before Last}}=E-\text{width}→\text{height of rectangle}=f\left(E-\text{width}\right)$