1. The problem statement, all variables and given/known data

Let us consider an approximation to an integral. Let f(x) be some continuous function on

[a, b]. We wish to find an approximation for the integral

I = int from a to b of f(x)dx

in the following manner:

Subdivide the interval into N intervals of length h = (b−a)/N. Let xi = ih for i = 0, . . . ,N.

Let

Ij = int from 0 to h of f(xj+t)dt

Find a cubic polynomial Pj (x) that goes through (xj , f(xj)), (xj + h/3,f(xj + h/3)),(xj+2h/3, f(xj+2h/3) and (xj+1,f(xj+1))

We form an approximation for the integral by letting

I=sum(j=0 to N-1) of w0*f(xj)+w1*f(xj+h/3)+w2*f(xj+2h/3)+w3*f(xj+1)

Find these weights, wi.

In 2 peices of code, plot the first three Bessel functions, J0(x), J1(x) and J2(x), on the

interval [0, 20]. The first peice of code should be a MATLAB function BJ(x, n) outputing

the approximation for the integral representation of Jn, given by

Jn(x) =(1/pi)int from 0 to pi of cos(nt − x sin t)dt

using the above method for 100 subdivisions of [0, pi]. The second peice of code should call

the function an produce the required plots with 2000 subdivisions of [0, 20].

Im just gobsmacked with this qn.. as i only started using matlab a couple of days, ago and have no programming experience.

what i have done so far is really no good, but i have no idea.

function [Jn]=BJ(x,n)

N=100;

b=pi;

a=0

h=(b-a)/N;

x=a:h:b;

xj=i*h;

Ij=0;

J0=cos(n*xj-x*sin(xj));

J1=cos(n*(xj+h/3)-x*sin(xj+h/3));

J2=cos(n*(xj+(2*h/3))-x*sin(xj+(2*h/3)));

J3=cos(n*(xj+1)-x*sin(xj+1));

for i=0:N;

Ij=Ij+w0*J0+w1*J1+w2*J2+w3*J3;

J=(1/pi)*Ij;

end

can someone please help me