Help with steps in this equation - formula for irradiance

[namendoza]

I need help showing how each step is done. Most important is the third step. Thanks for any advice or help in advance.

\(\displaystyle \displaystyle \phi=\int^\lambda_0 \Phi(\lambda) d\lambda=\int^\lambda_0 P(\lambda) \frac{\lambda}{hc} d\lambda\)

\(\displaystyle \displaystyle =\int^{1000nm}_{250nm} 1*10^9 \frac{\lambda*d\lambda}{6.625*10^{-34}*3*10^8} + \int^{2000nm}_{1000nm} 0.25*10^9 \frac{\lambda*d\lambda}{6.625*10^{-34}*3*10^8}\)


\(\displaystyle \displaystyle =\frac{1}{6.625*10^{-34}*3*10^8}\large{(}\normalsize{1}*10^9\frac{(1000*10^{-9})^2-(250*10^{-9})^2}{2}+0.25*10^9\frac{(2000*10^{-9})^2-(1000*10^{-9})^2}{2}\large{)}\)


\(\displaystyle \displaystyle =4.2*10^{21} \,m^{-2}s^{-1}\)

 

[Deleted member 4993]

I need help showing how each step is done. Most important is the third step. Thanks for any advice or help in advance.

\(\displaystyle \displaystyle \phi=\int^\lambda_0 \Phi(\lambda) d\lambda = \int^\lambda_0 P(\lambda) \frac{\lambda}{hc} d\lambda\)

\(\displaystyle \displaystyle =\int^{1000nm}_{250nm} 1*10^9 \frac{\lambda*d\lambda}{6.625*10^{-34}*3*10^8} + \int^{2000nm}_{1000nm} 0.25*10^9 \frac{\lambda*d\lambda}{6.625*10^{-34}*3*10^8}\)


\(\displaystyle \displaystyle =\frac{1}{6.625*10^{-34}*3*10^8}\large{(}\normalsize{1}*10^9\frac{(1000*10^{-9})^2-(250*10^{-9})^2}{2}+0.25*10^9\frac{(2000*10^{-9})^2-(1000*10^{-9})^2}{2}\large{)}\)


\(\displaystyle \displaystyle =4.2*10^{21} \,m^{-2}s^{-1}\)

Line (2) was integrated using:

\(\displaystyle \displaystyle \int_a^b(\lambda) d\lambda \ = \ \dfrac{b^2 - a^2}{2} \)

rest is algebra....

 

[namendoza]

Thank you so much. This helps quite a bit!