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# Numerical Approximation of Integrals

Whether you want to program a calculator or just approximation an integral on paper, break out some numerical approximation tools like Simpson's Rule and the Trapezoid Rule.

# Integral Approximation - Simpson's Rule

Using Simpson's rule, approximate $$\displaystyle{\int_0^1(5x^2+1)dx}.$$

What is the error when approximating $$\displaystyle{\int_0^{6}(6x^2+2x-3)dx}$$ using Simpson's rule?

Approximate $$\displaystyle{\int_{0}^{6}(x^3-x^2-7x-11)dx}$$ using Simpson's rule.

Sam and Lisa are approximating the definite integral $$\displaystyle{\int_0^{4}(3x^2+2x+1)dx}.$$ Sam used the right Riemann sum with $$4$$ intervals, and Lisa used Simpson's rule. If Sam's approximation is $$S$$ and Lisa's approximation is $$L,$$ what is $$S-L?$$

Given only the following three values of the function $$f$$: $f(0)=18, f(50)=14, f(100)=-5,$ approximate the integral $$\displaystyle{\int_0^{100}f(x)dx}$$ using Simpson's rule.

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