7-16= Use (a) the Trapezoidal Rule, (b) the Midpoint Rule,...

$7-16=$ Use (a) the Trapezoidal Rule, (b) the Midpoint Rule,
and (c) Simpson's Rule to approximate the given integral with
the specified value of $n .$ (Round your answers to six decimal
places.)
$$\int_{1}^{2} \sqrt{x^{3}-1} d x, \quad n=10$$

Key Concepts

Numerical Integration

Numerical integration is the process of approximating the value of a definite integral when an analytic solution is difficult or impossible to obtain. It involves estimating the area under a curve by dividing it into subintervals and summing the areas of simple geometric shapes defined over these intervals.

Composite Rules

Composite rules refer to the technique of applying a numerical integration method repeatedly over several subintervals to improve the accuracy of the approximation. By splitting the interval into n subintervals, one obtains a sum of approximations over each subinterval, thereby leading to a more precise estimate of the integral.

Trapezoidal Rule

The Trapezoidal Rule approximates the area under a curve by replacing the curve with a series of trapezoids. Each trapezoid’s area is computed using the average of the function values at the endpoints of the subinterval, yielding a linear approximation of the integral over that subinterval.

Midpoint Rule

The Midpoint Rule approximates the area under a curve by evaluating the function at the midpoint of each subinterval and multiplying by the subinterval’s width. This method often provides a better approximation than the Trapezoidal Rule for functions that are approximately linear over small intervals.

Simpson's Rule

Simpson's Rule is a numerical integration technique that uses parabolic arcs to approximate the function over each pair of subintervals. By fitting a quadratic polynomial through three consecutive points, Simpson's Rule typically provides a more accurate estimate than both the Trapezoidal and Midpoint Rules, especially for smooth functions.

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