Blog: A- Find the Riemann sum for f(x) = 7 sin x, 0 ≤ x ≤ 3π/2, with six terms, taking the sample points to be right endpoints. (Round your...

Wednesday, July 27, 2016

A- Find the Riemann sum for f(x) = 7 sin x, 0 ≤ x ≤ 3π/2, with six terms, taking the sample points to be right endpoints. (Round your...

Hello!

Summands of the Riemann sum have a form

$\displaystyle\delta{x}_{{n}}\cdot{f{{\left({x}_{{n}}\right)}}},$

where segments $\displaystyle{d}_{{n}}$ of length $\displaystyle\delta{x}_{{n}}$ cover the integration segment and each $\displaystyle{x}_{{n}}$ is in $\displaystyle{d}_{{n}}.$

In our problem all length of subintervals are considered equal, i.e.

$\displaystyle\delta{x}=\delta{x}_{{n}}=\frac{{\frac{{{3}\pi}}{{2}}}}{{6}}=\frac{\pi}{{4}}.$

$\displaystyle{x}_{{n}}$ 's are right endpoints, the formula for them is obviously

$\displaystyle{x}_{{n}}={n}\cdot\frac{\pi}{{4}},$ $\displaystyle{n}$ is from 1 to 6.

So the sum is $\displaystyle\frac{\pi}{{4}}{\sum_{{{n}={1}}}^{{6}}}{\left({7}{\sin{{\left({n}\cdot\frac{\pi}{{4}}\right)}}}\right)}.$

All terms are known, and this sum is equal to

$\displaystyle\frac{{{7}\pi}}{{4}}{\left(\frac{\sqrt{{{2}}}}{{2}}+{1}+\frac{\sqrt{{{2}}}}{{2}}+{0}+{\left(-\frac{\sqrt{{{2}}}}{{2}}\right)}+{\left(-{1}\right)}\right)}=\frac{{{7}\sqrt{{{2}}}}}{{8}}\pi.$

The approximate value is 3.887523.

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Wednesday, July 27, 2016

A- Find the Riemann sum for f(x) = 7 sin x, 0 ≤ x ≤ 3π/2, with six terms, taking the sample points to be right endpoints. (Round your...

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