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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.

Riemann Sums and Definite Integrals


Find the approximate value of \(\displaystyle{\int_0^{2} 7x^{2}dx}\) using a right Riemann sum by dividing the interval into \(4\) pieces.

Which of the following represents the approximation of \(\displaystyle{\int_{0}^{4}x^{5}dx}\) using a left Riemann sum?

Find the approximate value of \(\displaystyle{\int_0^1 (6x^2+2)dx}\) using a right Riemann sum, by dividing the interval into \(7\) parts.

The following is Alex's approximation of an integration by using a right Riemann sum: \[\frac{9}{5}\cdot\left(\left(\frac{3}{5}\right)^{7}+\left(\frac{6}{5}\right)^{7}+\left(\frac{9}{5}\right)^{7}+\left(\frac{12}{5}\right)^{7}+\left(\frac{15}{5}\right)^{7}\right).\] Which of the following integrals is Alex approximating?

What is the Riemann sum of the function \(f(x)= x^3-6x\) is in the interval \( [0, 6] \), if we divide it into 3 equal parts and use the midpoint of each interval?


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