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Simpson's rule and the Trapezoidal rule are approximations of the definite integral of a function. If the definite integral `int_a^b f(x)` has to be determined, the expression to approximate the value of the integral is given by the two rules as follows:
Simpson's rule: `int_a^b f(x) = ((b-a)/6)*(f(a) + 4*f((a+b)/2)+f(b))`
Trapezoidal rule: `int_a^b f(x) = (b-a)*(f(a)+f(b))/2`
The Trapezoidal rule determines the area under the graph by approximating it to that of a trapezoid. Simpson's rule approximates the area to that under a quadratic polynomial. This makes it a more complex method and one that yields a value closer to the definite integral that is actually being determined.
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