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In Matlab I am trying to use the composite Simpson's rule to find ##x_l## so that

$$170=\int^{x_l}_0 \sqrt{1+(y')^2} dx = \int^{x_l}_0 \sqrt{1+\left( \frac{x^2}{68000} \right)^2} dx $$

For convenience this can be written as

$$I(x) = 170 - \int^x_0 \sqrt{1 + (\frac{x^2}{68000})} dx$$

The limits of integration would be from ##0## to ##x=170##. Now to find the zero of this function I want to employ the bisection method while using Simpsons rule to evaluate the integral involved in evaluating ##I(x)## at each step.

Here's my code so far:

But I get lots of errors and the code does not run. I think it is because I need to get the Simpson's rule to calculate new value for the function at each midpoint of the bisection method. I'm not really sure how to fix this.

Alternatively, if bisection does not work with Simpson's method, I would appreciate it if anyone could show me how to exactly incorporate built in root finders into my code.

Any help would be greatly appreicated.

$$170=\int^{x_l}_0 \sqrt{1+(y')^2} dx = \int^{x_l}_0 \sqrt{1+\left( \frac{x^2}{68000} \right)^2} dx $$

For convenience this can be written as

$$I(x) = 170 - \int^x_0 \sqrt{1 + (\frac{x^2}{68000})} dx$$

The limits of integration would be from ##0## to ##x=170##. Now to find the zero of this function I want to employ the bisection method while using Simpsons rule to evaluate the integral involved in evaluating ##I(x)## at each step.

Here's my code so far:

Code:

```
a=0; b=170;
for x=[0:]
f = sqrt(1+((x.^2)./68000).^2);
%Simpson's rule
if numel(f)>1
n=numel(f)-1; h=(b-a)/n;
I= h/3*(f(1)+2*sum(f(3:2:end-2))+4*sum(f(2:2:end))+f(end));
else
h=(b-a)/n; xi=a:h:b;
I= h/3*(f(xi(1))+2*sum(f(xi(3:2:end-2)))+4*sum(f(xi(2:2:end)))+f(xi(end)));
end
f = 170 - I;
tol=1e-6;
while abs(b-a) > tol
x = (a+b)/2;
y = ff(x);
if y == 0
rv = x;
break
end
if ff(a)*y < 0
b = x;
else
a = x;
end
end
rv=(a+b)/2;
```

But I get lots of errors and the code does not run. I think it is because I need to get the Simpson's rule to calculate new value for the function at each midpoint of the bisection method. I'm not really sure how to fix this.

Alternatively, if bisection does not work with Simpson's method, I would appreciate it if anyone could show me how to exactly incorporate built in root finders into my code.

Any help would be greatly appreicated.

Last edited: