1. (15 points) (Numerical Integration) Recall that Newton-Cotes Quadrature approximate the integrand with a polynomial interpolant on equally-spaced nodes in an interval. Consider approximating the intergral \int_0^2 x^2 \sin(-x) dx (a) Consider using the Trapezoid rule and calculated the maximum approximation error. (b) Consider using the Simpson's rule and calculated the maximum approximation error. (c) Using smaller integration interval can reduce the approximation error. Consider divid- ing the interval [0, 2] into n sub-intervals and apply the quadrature rule to each interval. Find the number of sub-intervals required to approximate the following integrals with maximum approximation error upper bounded by $10^{-6}$ using the Trapezoid rule and Simpson's rule.
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Step 1: The Trapezoid rule for approximating the integral of a function f(x) over the interval [a, b] is given by: ∫[a, b] f(x) dx ≈ (b - a) * (f(a) + f(b)) / 2 The maximum approximation error for the Trapezoid rule is given by the formula: E ≤ K * (b - a)^3 / 12 Show more…
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For the integral of problem #3: (a) Use the Error Bound to find the bound for the error. (b) Compute the error made when using this estimate. 5. Use the Error Bound formula for the Trapezoidal Rule to determine N so that if ∫₀¹⁰ e⁻²ˣ dx is approximated using the Trapezoidal Rule with N subintervals, the error is guaranteed to be less than 10⁻⁴. 6. Estimate ∫₁⁵ ln x dx using the Trapezoidal Rule with n = 6 subintervals. 7. It is a fact, which you can take on faith, that the fourth derivative of f(x) = √(1 - cos²x/4) is always less than 2 (in absolute value). Determine N so that if ∫₀ᵭ/² √(1 - cos²x/4) dx is approximated using Simpson's Rule with N subintervals, the error will be less than 10⁻⁵. Remember that Simpson's Rule requires an even number of subintervals.
Adi S.
Use the error formulas to find n such that the error in the approximation of the definite integral is less than 0.0001 using the Trapezoidal Rule and Simpson's Rule. (In each case, choose n to be the smallest integer that satisfies these conditions.) (a) Trapezoidal Rule (b) Simpson's Rule
Consider the following integrals and the given values of $n .$ a. Find the Trapezoid Rule approximations to the integral using n anc 2n subintervals. b. Find the Simpson's Rule approximation to the integral using $2 n$ subintervals. It is easiest to obtain Simpson's Rule approximations from the Trapezoid Rule approximations, as in Example 6 c. Compute the absolute errors in the Trapezoid Rule and Simpson's $$ \int_{1}^{e} \frac{1}{x} d x ; n=50 $$
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Numerical Integration
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