1. ## Finding Riemann sums

The question:

An electrical signal S(t) has its amplitude |S(t)| tested (sampled) every 1/10 of a second. It is desired to estimate the energy over a period of half a second, given exactly by:

$\displaystyle (\int_0^\frac{1}{2} \! |S(t)|^2 \, \mathrm{d}t)^{\frac{1}{2}}$

The result of the measurement are shown in the following table:
$\displaystyle \begin{center} \begin{tabular}{| l | c | c| c| c| c|} \hline t & .1 & .2 & .3 & .4 & .5\\ \hline \| S(t) \| & 60 & 50 & 50 & 45 & 55 \\ \hline e(t) & 5 & 3 & 7 & 4 & 10 \\ \hline \end{tabular} \end{center}$

a) Using the above data for S(t), set up the appropriate Riemann sum and compute an appropriate value for the energy.

My attempt:
The set P (set of partitions) is clearly equal to {0, 1/10, 1/5, 3/10, 4/10, 1/2} so dt is 1/10 i.e. the width of each rectangle is 1/10.

My problem is with producing lower and higher Riemann sums. I'm used to having a function defined as something like $\displaystyle x^2$ and producing a general sum given dx (in this case, dt). In this question, we don't know the function, we're just given values.

Does this mean I take the max/min of each interval (e.g. [0, 1/10], [1/10, 1/5] etc.) and work out the partitions by hand? I'm a bit confused.

Any help would be greatly appreciated!

2. Originally Posted by Glitch
The question:

An electrical signal S(t) has its amplitude |S(t)| tested (sampled) every 1/10 of a second. It is desired to estimate the energy over a period of half a second, given exactly by:

$\displaystyle (\int_0^\frac{1}{2} \! |S(t)|^2 \, \mathrm{d}t)^{\frac{1}{2}}$

The result of the measurement are shown in the following table:
$\displaystyle \begin{center} \begin{tabular}{| l | c | c| c| c| c|} \hline t & .1 & .2 & .3 & .4 & .5\\ \hline \| S(t) \| & 60 & 50 & 50 & 45 & 55 \\ \hline e(t) & 5 & 3 & 7 & 4 & 10 \\ \hline \end{tabular} \end{center}$

a) Using the above data for S(t), set up the appropriate Riemann sum and compute an appropriate value for the energy.

My attempt:
The set P (set of partitions) is clearly equal to {0, 1/10, 1/5, 3/10, 4/10, 1/2} so dt is 1/10 i.e. the width of each rectangle is 1/10.

My problem is with producing lower and higher Riemann sums. I'm used to having a function defined as something like $\displaystyle x^2$ and producing a general sum given dx (in this case, dt). In this question, we don't know the function, we're just given values.

Does this mean I take the max/min of each interval (e.g. [0, 1/10], [1/10, 1/5] etc.) and work out the partitions by hand? I'm a bit confused.

Any help would be greatly appreciated!
left sum ... L = 0.1[S(0) + S(.1) + S(.2) + S(.3) + S(.4)]

right sum ... R = 0.1[S(.1) + S(.2) + S(.3) + S(.4) + S(.5)]

the table does not give a value for S(0), so it looks as though you can only compute the right sum.

3. Inside each interval you are given only two values- the values at the two endpoints. For the "lower sum" use the smaller of those two, for the "higher sum" use the larger.

In the first interval, from 0.1 to 0.2, you are given exactly two values- 60 and 50. The "lower sum" uses the smaller of those, 50, and the "higher sum" uses the higher, 60. Between .2 and .3 your two values are 50 and 50. Since those are the same, use 50 for both sums. Between .3 and .4, the two values are 50 and 45. Use 45 for the "lower sum" and 50 for the "higher sum". Finally, between .4 and .5, the two values are 45 and 55. Use 45 for the lower sum and 55 for the higher sum.

Summarizing, for the "lower sum" use 50, 50, 45, and 45. For the "higher sum" use 60, 50, 50, and 55.

4. For the lower sum I used: 0, 50, 50, 45, 45
I squared each (as per the function), and multiplied them by 1/10 to produce the area of each partition. My result was 905.

For the higher sum I used: 60, 60, 50, 50, 55
Again, I squared each, and multiplied by 1/10. My result was 1522.5

I took the average of the higher and lower sums, to get 1213.75. As per the question, the integral is to the power of 1/2, so I took the square root and got 34.84. However, the solution in my text is 36.95 (or square root of 1365).

I'm not sure where I went wrong. My answer is close, but incorrect. :/

Edit: I just realised that using '0' for the lower sum isn't correct. I used 60 instead, but I get 1393.75 which is slightly off the given solution.