Limit of Summation and Integral Solution Verification

[azatkgz]

Can someone check this solution.

Homework Statement

$$\lim_{n\rightarrow\infty}\sum^{n}_{i=1}\sqrt{\frac{1}{n^2}+\frac{2i}{n^3}}$$

The Attempt at a Solution

$$=\lim_{n\rightarrow\infty}\frac{1}{n}\sum^{n}_{i=1}\sqrt{1+\frac{2i}{n}}=\int^{1}_{0}\sqrt{1+2x}dx$$
for u=1+2x->du=2dx
$$\int^{1}_{0}\frac{\sqrt{u}du}{2}=\frac{1}{3}$$

 

[Gib Z]

That is correct.

 

[azatkgz]

My prof.says that I forgot to change something,while changing variables.

 

[Gib Z]

O perhaps the bounds on the integral? Sorry I am having a bad day..

 

[azatkgz]

yes, I think problem is on Integral.

 

[Gib Z]

I am sure of it now, you forgot the change the bounds. New limits of integration should be 3 and 1.

 

[azatkgz]

and the answer is
$$\int^{3}_{1}\sqrt{1+2x}dx=\frac{3^{\frac{3}{2}}-1}{3}$$
yes?

 

[Gib Z]

Correct, though it may look more pretty as $$\frac{3\sqrt{3}-1}{3}$$ lol.

 

[azatkgz]

Can you explain, please,why we must choose 3 and 1.

 

[Gib Z]

O simple. What the limits on the original integral mean are 'sum for values of x between 1 and 0'. You made the substitution u= 2x+1. So when the original integral says sum for x between 1 and 0, the new integral, where u is the new variable, must say 'sum for x between 1 and 0, and since u=2x+1, sum for u between 3 and 1'.

 

[azatkgz]

ok,thanks