Area summation problem under a curve

[Karol]

Homework Statement

Why, in:
$$\frac{\sqrt{1}+\sqrt{2}+...+\sqrt{n}}{n^{3/2}}$$
There is $~n^{3/2}$ in the denominator?

Homework Equations

The Attempt at a Solution

it should be:
$$S_n=\sqrt{c_1}\Delta x+\sqrt{c_2}\Delta x+...=\Delta x\cdot \sqrt{\Delta x}+\Delta x\cdot \sqrt{2\Delta x}+...=\sqrt{\Delta x}\cdot \Delta x(\sqrt{1}+\sqrt{2}+...+\sqrt{n})$$

 

[fresh_42]

Homework Statement

View attachment 135725
Why, in:
$$\frac{\sqrt{1}+\sqrt{2}+...+\sqrt{n}}{n^{3/2}}$$
There is $~n^{3/2}$ in the denominator?

Homework Equations

View attachment 135842

The Attempt at a Solution

it should be:
$$S_n=\sqrt{c_1}\Delta x+\sqrt{c_2}\Delta x+...=\Delta x\cdot \sqrt{\Delta x}+\Delta x\cdot \sqrt{2\Delta x}+...=\sqrt{\Delta x}\cdot \Delta x(\sqrt{1}+\sqrt{2}+...+\sqrt{n})$$

It should be $\sum_k f(c_k) \Delta x$. Now $c_k=k \cdot \Delta x\, , \,f(c_k)=\sqrt{c_k}$ and $\Delta x=\frac{1}{n}$. You simply stopped too soon before substituting $\Delta x=\frac{1}{n}$.

 

[Karol]

Thank you fresh_42