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Approximate the improper integral $$\int^{\frac {\pi}4}_0 \frac {\cos x~dx}{\sqrt{x}}$$ in these ways:

(a) using Riemann Sums

(b) First make the substitution $u$ $=$ ${\sqrt{x}}$ and then use Riemann sums

Hi! I was just reading a little about Riemann Sums and learning what improper integrals are. Is it possible that we can approximate this. I spoke to some people and they said know but one said yes.

Is it ok if you explain the working for me. I know I am teaching this to myself but your explaining would help. I would like to get head start on this topic. Please can you help me. Thanks!

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  • $\begingroup$ (b) is uneventful. With (a) we cannot use a left Riemann sum, but we can use a right Riemann sum. It will give a quite bad approximation. We can do better by using quite unequal intervals. But still not well. $\endgroup$ – André Nicolas Sep 5 '13 at 4:57

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