Estimation of summation

Question

A question told me to estimate the summation of $$\sqrt i$$ from 0 to 10000000 using integrals $$\sum_{i=0}^{10000000}\sqrt{i}=?$$ Is it okay to estimate by integrating it from 1 to 10000000, or am I supposed to use other methods like Riemann's Sum or?

Answer 1

If you let $f(x) = \sqrt{x}$, then you can think of the sum $$ \sum_{x=0}^{10\,000\,000} \sqrt{x}$$ as the area under the curve of the function $f(x)$ as $x$ ranges from $0$ to $10\,000\,000$.

So, we find that \begin{align*} \sum_{x=0}^{10\,000\,000}\sqrt{x} \approx \int_0^{10\,000\,000} \sqrt{x}\,dx &= \frac23 x^{3/2}\bigg|_0^{10\,000\,000}\\ &= \frac{20\,000\,000\,000 \sqrt{10}}3\\ &\approx 2.1082 \times 10^{10}. \end{align*} In fact, this approximation is pretty good since $$ \sum_{n=0}^{10\,000\,000} \sqrt{n} = 2.10818 \times 10^{10}.$$