Use Simpson's rule with the number of strips specified to approximate the following definite integrals: (a) \(\int_{0}^{0.8} \tan ^{2} x \mathrm{~d} x, 8\) strips (b) \(\int_{1}^{2} \sqrt{1+x^{3}} \mathrm{~d} x, 10\) strips

When working with calculus, one often comes across the need to calculate the area under a curve, or more technically, to evaluate a definite integral. While some functions allow for exact integration, many practical problems involve functions that are difficult or even impossible to integrate analytically. This is where the definite integrals approximation comes into play. Instead of finding a precise area, approximation methods, such as Simpson's rule, estimate the value by summing up the areas of shapes (typically rectangles, trapezoids, or parabolas) that are easier to calculate.

Simpson's rule is particularly useful because it provides a better approximation than many other numerical methods by using parabolic arcs instead of straight lines. The rule is applied by dividing the integration range into equal sections (strips) and then calculating the area under parabolas that pass through the function's values at specific points along the interval. To ensure accuracy, especially for functions with high rates of change, using more strips is recommended. It's the balance between computational effort and the desired accuracy that guides the choice of how many strips to use for the approximation.

Numerical integration consists of a range of algorithms for calculating the numerical value of a definite integral. The basic idea is to compute the accumulation of a quantity whose rate of change is given by a function. For example, integrating a velocity function gives us the total distance traveled. Simpson's rule, along with other techniques like the Trapezoidal rule and Midpoint rule, falls under numerical integration methods known as quadrature.

Understanding the concept of quadrature is vital for calculus for engineers, who often use these numerical methods to solve real-world problems where exact solutions are unfeasible. The efficiency of Simpson's rule lies in its ability to closely approximate the area beneath curves that are smooth or well-behaved over the interval without requiring the function's antiderivative. This efficiency makes it a preferred method in many engineering applications such as finding the center of mass, the moment of inertia, or assessing the total energy of a system when only empirical data is available.

Calculus is the mathematical study of continuous change, and it is a fundamental language in engineering. Calculus for engineers involves applying concepts of differentiation and integration to solve engineering problems. Engineers often encounter complex systems where exact solutions are not easily obtained; numerical methods like Simpson's rule provide practical tools to tackle such scenarios. The advantage for engineers using Simpson's rule lies in its straightforward application and its ability to yield precise results without complex mathematical operations.

As an engineer, you may not always need the exact area under a curve. Instead, a reliable approximation might satisfy design specifications or provide enough information for the analysis of physical systems. Numerical integration techniques like Simpson's rule are therefore indispensable in engineering fields, ranging from civil to aerospace engineering, where they are used in the design, analysis, and simulation of various processes and systems.