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Section 1-10 : Approximating Definite Integrals

For each of the following integrals use the given value of n to approximate the value of the definite integral using

  1. the Midpoint Rule,
  2. the Trapezoid Rule, and
  3. Simpson’s Rule.

Use at least 6 decimal places of accuracy for your work.

  1. \( \displaystyle \int_{{ - 2}}^{4}{{\sin \left( {{x^2} + 2} \right)\,dx}}\) using \(n = 6\)
  2. \( \displaystyle \int_{0}^{4}{{\sqrt[3]{{{x^4} + 6}}\,dx}}\) using \(n = 6\)
  3. \( \displaystyle \int_{1}^{5}{{{{\bf{e}}^{\cos \left( x \right)}}\,dx}}\) using \(n = 8\)
  4. \( \displaystyle \int_{3}^{5}{{\frac{1}{{1 - \ln \left( x \right)}}\,dx}}\) using \(n = 6\)
  5. \( \displaystyle \int_{{ - 3}}^{1}{{\sin \left( x \right)\cos \left( {{x^2}} \right)\,dx}}\) using \(n = 8\)