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A blackbody is also a perfect emitter giving off electromagnetic waves at all frequencies. A detector could measure the intensity of the radiation it receives through the prism. By moving the detector to different positions, you could measure the intensity of light as a function of color or wavelength.

So each wavelength has a finite intensity? If we consider wavelengths in the interval $[\lambda, \lambda + d\lambda]$ and each wavelength in this interval has a finite intensity, then the total intensity for this interval would be infinite because this interval has infinitely many wavelengths. Where am I going wrong? Are there not infinitely many wavelengths within this interval?

Between the red and green beams there are infinitely many beams with a finite intensity, so the total for this interval would be infinite?

Between the red and green beams there are infinitely many beams with a finite intensity, so the total for this interval would be infinite?

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This is a plot of the energy content Iintensity) versus wavelength of a black body radiator:

black body

How is this plot made? By using the Planck radiation formula. This means that a point is drawn with a width Δ(λ) small enough that the plot appears continuous and not a histogram. If the interval were large, it would be clear that the height was the sum of all the energy intensities for those wavelengths under that interval. This still is true no matter how small the Δ(λ), so there is no problem with infinities .

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No, for the same reason that the integral of every continuous function f(x) from $x_0$ to $x_0+\Delta x$ isn't infinite.

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