Black Body Radiation (Awkward integral)

[CanIExplore]

Homework Statement

What percentage of the Sun’s blackbody radiation spectrum falls into the visible light spectrum (400-700 nm). Where T=5000K
Hint: Integrate over frequencies

Homework Equations

B=2h$$\nu$$³c^-2 (e^h$$\nu$$/kT-1)^-1

Where $$\nu$$ is the frequency of the light.

The Attempt at a Solution

Ok so the problem is very straightforward, I'm just having trouble evaluating the integral. I need to integrate the equation for the brightness (B) over the frequency $$\nu$$ where the limits are given by the span of wavelengths in the visible part of the spectrum. The integral then just looks something like this:
$$\int B$$=$$\int \nu$$³(e^$$\nu$$-1)^-1 , where i excluded the constants.

I tried integration by parts but it didn't work. I also plugged it into mathematica and got a very weird, long answer that didn't make sense. When I asked my TA about it he told me I had to solve it numerically. What does it mean to solve an integral numerically? Am I just supposed to plug in the lower limit and then subtract that from what I get when I plug in the upper limit? I am thinking of the fundamental theorem of calculus here.

Thanks

 

[Cyosis]

This integral doesn't have a primitive in terms of elementary functions. To evaluate this integral you can use the Raleigh-Jeans approximation by looking it up in your book or by using the Taylor series of the exponent up to first order.

 

[phyzguy]

When you integrate a function numerically, you basically divide the area up into a bunch of narrow rectangles or trapezoids and add them all up. There are a lot of ways to do this, try reading this:

http://en.wikipedia.org/wiki/Numerical_integration

To accomplish this, you could write a program to do it, or use a canned program. Mathematica has a function NIntegrate which will do numerical integration and, given the function and the endpoints, will just return a number.

 

[CanIExplore]

Thanks guys