Writing the value of an integral as a function of one of the integrated variables?

[BetterSense]

This is not homework. I just need help tackling this analytical problem which is apparently beyond my skills.

I am modeling an imaginary physical situation. What I have here in my brain, is a photo sensor in a dark room, pointed squarely at a window, from some distance away. Actually the window is a diffuse hemispheric dome, 'doming' outward from the room. It is perfectly diffuse, so all parts of it are the same brightness. I need to figure out how close to the window I should hold the sensor, to achieve the maximum reading from the sensor.*

I consider the effect on the sensor of an infinitesimal area element of the dome, so that I can integrate over the whole dome later. The light falling on the sensor from such an element of the dome falls off as a function of the distance from said element, and as a function of the angle between that element and the center axis of the sensor (the sensor itself's sensitivity is a function of acceptance angle). I have already done all this, and this is not the problem.

The problem is that I can integrate over the whole dome at any given distance from the window and obtain a "brightness factor" for that distance from the window. Doing so involves integrating over a range of distances equal to the radius of the hemispherical window, though. Now, what I need is that (calculable for any distance!) brightness factor as a function of distance from the window. Basically what I need to do is do that integral over the dome surface for every distance from touching the inside of the hemispheric window to infinity. And I don't know how to do that.

In real life what I would do is use a spreadsheet to calculate the integral for a 'lot' of 'closely-spaced' distances, make an XY plot, and pick what looks like the peak. However, I want to be able to do this analytically. Help?


*The answer is not "as close as possible" because the sensor's sensitivity is a function of the angle so as the sensor approaches and intrudes into the hemisphere, less of the hemisphere's surface can be "visible" to the sensor.

 

[torquil]

First of all, your end result will not depend on any specific value of one of the integrated variables. That doesn't make sense. You should not be integrating of the variable that represents the sensor's distance L to the window. L will enter into the calculation as a adjustable parameter, not an integration variable. So L will enter into the calculation, but you should only integrate over two angular coordinates that span the solid angle of the window.

The problem is that I can integrate over the whole dome at any given distance from the window and obtain a "brightness factor" for that distance from the window.

From this statement it seems to me that you have already done what you wanted? You have done it for "any given distance" L, so you already the expression as a function of L? I probably misunderstood something here, though.

Or did you mean by this that you have done the analytical calculation without taking into account the variable angular sensitivity of the detector?

Torquil

 

[BetterSense]

From this statement it seems to me that you have already done what you wanted?

Almost! This is very useful because now I can calculate the brightness factor at any distance y' by doing an integral over a certain range of y-values of the effect those y-values have on my point of interest y'. To solve the problem in the real world I could just us a computer program to calculate that integral at a lot of distances y', and then fit a curve to it. But I want to actually write the brightness factor as a function of distance y'. Then I can take the derivative of that and set it equal to zero to analytically show the maximum. How can I do that?

You have done it for "any given distance" L, so you already the expression as a function of L?

I have done the first part, but not the second! What I have is an algorithm for finding B at any distance y:

To obtain the brightness factor B for any point y', go to point y' and integrate the effect of light on that point from the range y=a to y=b

but since this involves doing an integral over a range of y-values, I can't write this as a function of y. I end up with int(y=a->y=b)(f(y)dy)=B. This let's me find B for any y' but how can I write it as a function of y when it involves integrating over a range of y? It might be possible to do so, but I don't know how to.