# numerical methods engineering

1)Consider the expression for blackbody radiation given by Equation 4.62 . The integral of this expression over all wavelengths, that is, , gives the total energy radiated by a blackbody per unit area and time. Using Simpson’s rule, compute this integral at T=1000K as accurately as possible. The analytical result is given in the literature as , where is known as the Stehan-Boltzman constant and has a value of 5.67×10^(-8) W/m^2K^4. Compare your numerical result with the analytical value at 1000K. Equation 4.62 is also attached as an image.

2)The RMS value of an electric current I(t), where I varies periodically with time t is given by the expression where is the time period for one cycle in the variation of I(t). If I(t) is given as 5e^(-t)sin4pit, with =0.5 seconds, compute the RMS value using Simpson’s Rule.

r(cm) |
0 |
0.2 |
0.5 |
0.6 |
0.8 |
0.9 |
1.0 |

V(cm/s) |
1.0 |
0.96 |
0.75 |
0.64 |
0.36 |
0.19 |
0.0 |

The volume flow in the pipe is given by the integral

, where R is the radius of the pipe. Using the data given, compute the integral.

4)Using Simpson’s rule, compute the improper integral as accurately as possible.