The intensity of light with wavelength λ traveling through a diffraction grating with N slits at an angle θ is given by I(θ) = N2 sin2(k)/k2, where k = (πNd sin θ)/λ and d is the distance between adjacent slits. A helium-neon laser with wavelength λ = 632.8 × 10−9 m is emitting a narrow band of light, given by −10−6 < θ < 10−6, through a grating with 10,000 slits spaced 10−4 m apart. Use the Midpoint Rule with n = 10 to estimate the total light intensity.

[tex] \bf \int_{-10^{-6}}^{10^{-6}} I(\theta)d\theta=\int_{-10^{-6}}^{10{-6}}\frac{10^8sin^2(\frac{\pi 10^9sin\theta}{632.8})}{(\frac{\pi 10^9sin\theta}{632.8})^2}d\theta[/tex]

Since [tex] \bf I(\theta)[/tex] is an even function

[tex] \bf \int_{-10^{-6}}^{10^{-6}} I(\theta)d\theta=2\int_{0}^{10^{-6}}I(\theta)d\theta[/tex]

and we only have to divide the interval [tex] \bf [0,10^{-6}][/tex] in five equal sub-intervals [tex] \bf I_1,I_2,I_3,I_4,I_5[/tex] with midpoints [tex] \bf m_1,m_2,m_3,m_4,m_5[/tex]

The sub-intervals and their midpoints are

[tex] \bf I_1=[0,\frac{10^{-6}}{5}]\;,m_1=10^{-5}\\I_2=[\frac{10^{-6}}{5},2\frac{10^{-6}}{5}]\;,m_2=3*10^{-5}\\I_3=[2\frac{10^{-6}}{5},3\frac{10^{-6}}{5}]\;,m_3=5*10^{-5}\\I_4=[3\frac{10^{-6}}{5},4\frac{10^{-6}}{5}]\;,m_4=7*10^{-5}\\I_5=[4\frac{10^{-6}}{5},10^{-6}]\;,m_5=9*10^{-5}[/tex]

By the midpoint rule

[tex] \bf \int_{0}^{10^{-6}}I(\theta)d\theta\approx\frac{10^{-6}}{5}[I(m_1)+I(m_2)+...+I(m_5)][/tex]

computing the values of I:

[tex] \bf I(m_1)=I(10^{-5})=\frac{10^8sin^2(\frac{\pi 10^9sin(10^{-5})}{632.8})}{(\frac{\pi 10^9sin(10^{-5})}{632.8})^2}=13681.31478[/tex]

[tex] \bf I(m_2)=I(3*10^{-5})=\frac{10^8sin^2(\frac{\pi 10^9sin(3*10^{-5})}{632.8})}{(\frac{\pi 10^9sin(3*10^{-5})}{632.8})^2}=4144.509447[/tex]

Similarly with the help of a calculator or spreadsheet we find

[tex] \bf I(m_3)=3.09562973\\I(m_4)=716.7480066\\I(m_5)=211.3187228[/tex]

and we have

[tex] \bf \int_{0}^{10^{-6}}I(\theta)d\theta\approx\frac{10^{-6}}{5}[I(m_1)+I(m_2)+...+I(m_5)]=\frac{10^{-6}}{5}(18756.98654)=0.003751395[/tex]

Finally the the total light intensity

would be 2*0.003751395 = 0.007502795