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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 - Trapezium Rule

The following is Devin's approximation of an integral using the Trapezium rule: $\frac{6}{5}\cdot\left(\left(\frac{3}{5}\right)^{5}+2\cdot\left(\frac{6}{5}\right)^{5}+2\cdot\left(\frac{9}{5}\right)^{5}+2\cdot\left(\frac{12}{5}\right)^{5}+\left(\frac{15}{5}\right)^{5}\right).$ Which of the following integrals is Devin approximating?

Find the approximate value of $$\displaystyle{\int_0^{2} 7x^{3}dx}$$ using the Trapezium rule, by dividing the interval into $$4$$ pieces.

What is the Trapezium rule approximation to the definite integral $$\displaystyle{\int_{-1}^1 (10-2x^2)dx}$$ using four intervals?

Which of the following represents the approximation of $$\displaystyle{\int_{0}^{2}x^{4}dx}$$ using the Trapezium rule?

Find the approximate value of $$\displaystyle{\int_0^1 (12x^2+2)dx}$$ using the Trapezium rule, by dividing the interval into $$5$$ parts.

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