The trapezoidal and Simpson's rules are methods of numerical approximations of the integral of a function.
The trapezoidal rule
The trapezoidal rule involves dividing the area under a function into trapezoids.
There is a set formula for the rule, requiring a number of subintervals that is divisible by 4.
The formula is defined as: b-a)/2n*[f(n)+2(f(n+1))+2(f(n+2))+2(f(n+3))....(f(n+x))].
Examples
1) S from 0 to 1 of sqrt(sin(x^2-1)) using 4 subintervals.
2) S from 1 to 1.5 of e^(3cos(2x)). 4 subintervals
3) S from 0 to 2 of x^3-7x^2+2x-4 using 8 subintervals.
There is a set formula for the rule, requiring a number of subintervals that is divisible by 4.
The formula is defined as: b-a)/2n*[f(n)+2(f(n+1))+2(f(n+2))+2(f(n+3))....(f(n+x))].
Examples
1) S from 0 to 1 of sqrt(sin(x^2-1)) using 4 subintervals.
2) S from 1 to 1.5 of e^(3cos(2x)). 4 subintervals
3) S from 0 to 2 of x^3-7x^2+2x-4 using 8 subintervals.
Simpson's Rule
Simpson's rule is similar to the trapezoidal rule, except it compares the function to a series of parabolic curves.
The formula to find simpson's rule is a)*[f(n)+2(f(n+1))+2(f(n+2))+2(f(n+3))....(f(n+x)).