- #1

Karol

- 1,380

- 22

## Homework Statement

Why specifically 1/2 is the coefficient in C

_{K}? the sum, basically, doesn't change except for the coefficient. i can choose it as i want.

I understand the sum must equal the integral but i guess that's not the reason

## Homework Equations

Area under a curve as a sum:

$$S_n=f(c_1)\Delta x+f(c_2)\Delta x+...+f(c_n)\Delta x$$

## The Attempt at a Solution

$$f(c_1)\Delta x=\frac{x_k-x_{k-1}}{\frac{\sqrt{x_{k-1}}+\sqrt{x_k}}{2}}=...=2(\sqrt{x_k}-\sqrt{x_{k-1}})$$

$$S_n=2(\sqrt{x_1}-\sqrt{x_0}+\sqrt{x_2}-\sqrt{x_1}+...+\sqrt{x_n}-\sqrt{x_{n-1}})=2(\sqrt{x_n}-\sqrt{x_0})$$