Use the error formulas to find n such that the error in the...

Use the error formulas to find $n$ such that the error in the approximation of the definite integral is less than 0.0001 using (a) the Trapezoidal Rule and (b) Simpson's Rule.
$$
\int_{1}^{3} \frac{1}{x} d x
$$

Key Concepts

Error Analysis for Numerical Integration

This concept involves estimating and controlling the difference between the true value of an integral and its numerical approximation. It typically requires knowing the behavior of the function’s higher order derivatives, which are used to bound the error in numerical methods. The idea is to determine how many subintervals or what step size are needed for a given tolerance in the approximation error.

Trapezoidal Rule

The Trapezoidal Rule is a numerical integration method that approximates the area under a curve by dividing the integration interval into subintervals and replacing the function by a straight line that connects the endpoints of each subinterval. Its error analysis formula depends on the second derivative of the integrand, and it suggests that the error decreases quadratically with the number of subintervals, allowing us to choose an appropriate n to achieve a desired accuracy.

Simpson's Rule

Simpson's Rule is a numerical approximation technique that uses quadratic polynomials to approximate segments of the function over the interval. It generally provides a higher accuracy than the Trapezoidal Rule because it uses parabolic approximations rather than linear ones. The error formula for Simpson's Rule involves the fourth derivative of the function, and it shows that the error decreases at a faster rate with an increasing number of subintervals, permitting stricter error bounds with fewer intervals.

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