Use a computer algebra system and the error formulas to find...

Use a computer algebra system and the error formulas to find $n$ such that the error in the approximation of the definite integral is less than $0.00001$ using (a) the Trapezoidal Rule and (b) Simpson's Rule.
$$
\int_{0}^{2} \sqrt{1+x} d x
$$

Key Concepts

Numerical Integration

Numerical integration refers to the use of computational algorithms to approximate the value of definite integrals. This is especially useful when the antiderivative of the function is difficult or impossible to determine analytically. Various rules, such as the Trapezoidal and Simpson's rules, are employed to systematically estimate the area under a curve by breaking it into smaller regions.

Trapezoidal Rule

The Trapezoidal Rule approximates the area under a curve by dividing the integration interval into subintervals and approximating the function with straight line segments over each subinterval. The resulting trapezoids' areas are then summed to estimate the integral, offering a balance between simplicity and accuracy.

Trapezoidal Rule Error Bound

The error bound for the Trapezoidal Rule is typically expressed using the second derivative of the integrand. This bound helps determine how fine the partition (or how large the number of subintervals) should be to ensure that the absolute error in the integral approximation does not exceed a specified tolerance.

Simpson's Rule

Simpson's Rule improves upon the Trapezoidal Rule by using parabolic arcs instead of straight lines to approximate the function over each pair of subintervals. This typically provides a more accurate approximation by fitting the curvature of the function more closely, especially when the function is well-behaved.

Simpson's Rule Error Bound

The error bound for Simpson's Rule involves the fourth derivative of the function. This bound is used to ascertain the number of subintervals required so that the composite Simpson's approximation achieves an error below a pre-determined tolerance, ensuring a high degree of accuracy.

Error Analysis and Tolerance

Error analysis in numerical integration involves quantifying the difference between the exact value of the integral and its numerical approximation. By establishing error bounds through derivative estimates, one can determine the necessary fineness of the partition (or the number of subintervals) to guarantee that the approximation error remains within acceptable limits, as specified by a given tolerance.

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