numerical methods engineering

ccruz7105

1)Consider the expression for blackbody radiation given by Equation 4.62 $E_{b\lambda }(\lambda ,T)=C_{1}/(\lambda ^5)[e(^C2/\lambda T) -1]$. The integral of this expression over all wavelengths, that is, $\int_{0}^{infinity}E_{b\lambda }d\lambda$, 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 $\sigma T^4$, where $\sigma$ is known as the Stehan-Boltzman constant and has a value of 5.67x10^(-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 $I_{RMS}=\frac{1}{t_{c}}\sqrt{\int_{0}^{t_{c}}}I^2dt$ where  $t_{c}$ is the time period for one cycle in the variation of I(t). If I(t) is given as 5e^(-t)sin4pit, with $t_{c}$=0.5 seconds, compute the RMS value using Simpson's Rule.

3)The fluid velocity V is measured at several radial locations r for flow in a circular pipe of radius 1 cm. The velocities in cm/s are tabulated as follows
 r(cm) 0 0.2 0.5 0.6 0.8 0.9 1 V(cm/s) 1 0.96 0.75 0.64 0.36 0.19 0
The volume flow in the pipe is given by the integral $\int_{0}^{R}V(r)2\pi rdr$, where R is the radius of the pipe. Using the data given, compute the integral.

4)Using Simpson's rule, compute the improper integral $\int_{0}^{infinity}\frac{2dx}{1+e^-^x+x^2}$ as accurately as possible.

• Posted: 5 years ago
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