numerical methods engineering

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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

1.0

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 \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.

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