### Home > PC > Chapter 2 > Lesson 2.3.3 > Problem2-98

2-98.

Left = (width)(height of each rectangle)

$\text{Left} =(0.5)(\sqrt{3+2}^2+\sqrt{3.5+2}^2+\sqrt{4+2}^2+\sqrt{4.5+2}^2$

$f(x_{0})\ \ \ \ \ \ \ \ \ f(x_{1})\ \ \ \ \ \ \ \ f(x_{2})\ \ \ \ \ \ \ f(x_{3})$

Right = (width)(height of each rectangle)

$\text{Right} =(0.5)(\sqrt{3.5+2}^2+\sqrt{4+2}^2+\sqrt{4.5+2}^2+\sqrt{5+2}^2$

$f(x_{1})\ \ \ \ \ \ \ \ f(x_{2})\ \ \ \ \ \ \ f(x_{3})\ \ \ \ \ \ \ \ \ f(x_{4})$

$\text{Left endpoints} =\displaystyle\sum\limits_{k=0}^30.5\sqrt{(3+0.5k)^2+2}$

What modification results in the right endpoints?

The left endpoint indices go from k = 0 to k = 3 whereas the right endpoint indices go from k = 1 to k = 4.