Confused about if this is a Riemann sum.

Question

The question in its original form is as follows. $$\text {Find the value of } \lim_{n\to \infty} \sum _{k=0} ^{k=n} \left(\int _{\frac {k}{n}}^{\frac {k+1}{n}} \sqrt {(nx-k)(1-(nx-k))}dx\right) $$ .$$\text {Attempt } $$ After substituting $nx-k=u $ the problem changes to computing $$\lim_{n\to\infty}\sum _0^n \left(\int\sqrt {(u )(1-u)}du\right)\frac {1}{n}$$ Now I dont know what to do with this sum. Is this a Riemann sum so that the ultimate form is computing $$\int_0^1 \left(\int_0^1\sqrt {u(1-u)}du\right)du $$.

Answer 1

You have

$$\lim_{n\to\infty}\sum_{k=0}^n \left(\int_0^1\sqrt{u(1-u)}\,\mathrm du\right)\frac {1}{n}$$

Now set $$C:=\int_0^1\sqrt{u(1-u)}\,\mathrm du$$

Then

$$\lim_{n\to\infty}\sum_{k=0}^n \left(\int_0^1\sqrt{u(1-u)}\,\mathrm du\right)\frac {1}{n}=\lim_{n\to\infty}\frac1n\sum_{k=0}^n C=C\lim_{n\to\infty}\frac{n+1}n=C$$

Of course it can also be written as

$$\int_0^1 C\,\mathrm dx=C$$