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?
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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 $$ \sum_{n=0}^{10\,000\,000} \sqrt{n} = 2.10818 \times 10^{10}.$$
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Yes! I believe in this case, $x$ and $i$ are just placeholder variables. So it will apply. – Nov 08 '19 at 17:36
EDIT: It's $$\sum_{i=0}^{10000000}\sqrt{i}=?$$
@AndrewChin
– Jisbon Nov 08 '19 at 08:14