PART II: Numerical Integration

$9\mathrm{.}35$ The figure shows the output pulse from an MDS defibrilla$-$

tor. The voltage as a function time is given by:

$v(t)=3500sin(140t)e^{-63t}V$

The energy, $E,$ delivered by this pulse can be calculated by:

$E={\displaystyle \underset{0}{\overset{t}{}}}\frac{[v(t){]}^2}{R}dt$ Joules.

where $R$ is the impedance of the patient. For $R=50.$

Determine the energy, $E,$ from $t=0$ to $t=15$ milliseconds $($ ms $).$

a$.$ PROGRAMMING: Make a plot of the function $f(t)=(\frac{1}{R})v(t{)}^2$ vs$.$ time $(t)$ using Matlab.

b$.$ MANUAL PART: Determine the energy E using the method indicated below. For the first three methods, work with six $(6)$ subintervals. For Simpson's

$\frac{1}{3}$ work with eight $(8)$ subintervals, and Simpson's $\frac{3}{8}$ method work with nine $(9)$ subintervals. For each case, work with $(4)$ decimals. Compare the

results in a table.

Rectangle method $($left$).$

Midpoint method.

Trapezoidal method.

Simpson's $\frac{1}{3}$ method.

Simpson's $\frac{3}{8}$ method.

c$.$ PROGRAMMING: Implement all methods of integration in a Matlab program.

$9\mathrm{.}14$ The value of $$ can be calculated from the integral $=\frac{1}{2}{\displaystyle \underset{-1}{\overset{1}{}}}\frac{4}{1+x^2}dx.$

$($a$)$ Approximate $$ using the composite trapezoidal method with six subintervals.

$($b$)$ Approximate $$ using the composite Simpson's $\frac{1}{3}$ method with six subintervals.

$9\mathrm{.}14$ The value of $$ can be calculated from the integral $=\frac{1}{2}{\displaystyle \underset{-1}{\overset{1}{}}}\frac{4}{1+x^2}dx.$

$($a$)$ Approximate $$ using the composite trapezoidal method with six subintervals.

$($b$)$ Approximate $$ using the composite Simpson's $\frac{1}{3}$ method with six subintervals.