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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.

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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