# Thread: Definite integral divided by Definite Integral

1. ## Definite integral divided by Definite Integral

I am unsure how to solve this. I have values for all variables for each wavelength in 10 nm increments, 380,390,400....
My thought was to sum each value of $\displaystyle \tau$ at the given wavelength, but that is not correct.

$\displaystyle \tau_{SB} = \frac{\int_{380}^{500}\tau(\lambda) \times W_B(\lambda) d\lambda}{\int_{380}^{500} W_B(\lambda) d\lambda}$

Thanks for any help.

2. ## Re: Definite integral divided by Definite Integral

Hey bmccardle.

Are you trying to figure out some sort of average/expectation?

You should probably explain what you are trying to achieve here for further information.

3. ## Re: Definite integral divided by Definite Integral

This formula evaluates the transmission of a blue light, which is from 380 nm to 500 nm. The overall transmission for the filter I am trying to evaluate is 80%.
$\displaystyle W_B$ is a weighting factor at a certain wavelength and $\displaystyle \tau$ is the transmission at a particular wavelength. I thought you could sum up the value at each wavelength, but that is not correct.
Thanks for the help

The transmission at:
380 = 2.7
390 = 3.9
400 = 8.1
410 = 14.9
420 = 45.9
430 = 84.9
440 = 88.4
450 = 89.3
460 = 89.2
470 = 87.0
480 = 89.6
490 = 89.6
500 = 89.3

The weighting factor:

380 = 2
390 = 10
400 = 47
410 = 269
420 = 660
430 = 771
440 = 911
450 = 946
460 = 864
470 = 706
480 = 532
490 = 266
500 = 122

4. ## Re: Definite integral divided by Definite Integral

So, you have $\tau$ and $W_B$ at discrete points?

In which case, you will want to estimate the integrals. There are various methods to do this. The Trapezium Rule provides a good balance between simplicity and accuracy.

5. ## Re: Definite integral divided by Definite Integral

Would it be the sum of $\displaystyle \tau \times W_B$ divided by the sum of $\displaystyle W_B$? This is close to what the answer should be

4860.815/6106 = 0.7961 or 79.61%

6. ## Re: Definite integral divided by Definite Integral

If the weights add up to one [and are strictly positive] then what you are doing in the top integral is getting an expectation.

Also - how are the weights determined?

I assume you have a list of frequencies and you want to look at how much of a frequency is passed through relative to how much of some frequency is transmitted.

You should elaborate on what the counts are [and how you get them] along with how you determine your weightings for more information.