Question

Approximate the integral to three decimal places using the indicated rule. \(\int_{0.1}^{0.5} \frac{\cos x}{x} d x ;\) trapezoidal rule; \(n=4\)

Short answer

0.885

Step-by-step solution

Identify the Interval and Calculate Bin Width

The integration is over the interval from 0.1 to 0.5. We are using the trapezoidal rule with 4 intervals (\(n=4\)). First, calculate the width of each bin (\(h\)) using the formula:\[ h = \frac{b-a}{n} \]where \(a = 0.1\) and \(b = 0.5\). Therefore:\[ h = \frac{0.5 - 0.1}{4} = 0.1 \]

Determine the Function Values

Next, we calculate the function values at the endpoints and the intermediate points of the intervals. These points are: \(x_0 = 0.1\), \(x_1 = 0.2\), \(x_2 = 0.3\), \(x_3 = 0.4\), and \(x_4 = 0.5\).Calculate each function value:- \( f(x_0) = \frac{\cos(0.1)}{0.1} \)- \( f(x_1) = \frac{\cos(0.2)}{0.2} \)- \( f(x_2) = \frac{\cos(0.3)}{0.3} \)- \( f(x_3) = \frac{\cos(0.4)}{0.4} \)- \( f(x_4) = \frac{\cos(0.5)}{0.5} \)

Key concepts

Numerical Integration

Numerical Integration is a powerful tool in calculus used to estimate the value of integrals. Not all functions have antiderivatives that can easily be expressed in closed forms. When we face such situations, numerical integration methods come to the rescue. These methods provide us a way to approximate integrals even for complex functions or when exact integration is not feasible.

There are several methods of numerical integration, including the trapezoidal rule which is quite simple and effective. It involves dividing the area under the curve into a series of trapezoids rather than rectangles, which can lead to a more accurate approximation. Using the trapezoidal rule, particularly with more intervals, can enhance precision. This flexibility makes numerical integration very useful for both beginners and advanced users alike.

Approximate Integration

Approximate Integration refers to methods that provide an estimated value of a definite integral. It is essential when exact values are difficult to obtain. Approximating an integral becomes necessary with complex functions or on large intervals where traditional integration is computationally tedious.

The trapezoidal rule is one common technique for approximate integration. It uses straight-line segments to approximate the curve of the function across small intervals. By summing the areas of these trapezoids, we can estimate the total area under the curve. Though an approximation, with a higher number of smaller intervals, this method yields increasingly accurate results. As with anything in mathematics, understanding the balance between accuracy and computational efficiency is crucial when using approximate integration.

Calculus Applications

Calculus Applications are vast, ranging from physics to engineering, even in areas like economics and biology. Integral calculus, in particular, has critical applications whenever there is a need to calculate areas, volumes, or accumulations. For example, finding the area under velocity-time graphs, which gives displacement, is fundamental in physics.

Numerical integration and approximate integration methods, as discussed, are not mere academic exercises. They have real-world relevance, allowing practitioners to estimate integrals where exact solutions aren't possible. This capability is crucial in simulations, optimizations, and various predictive models used in technology and science. Understanding and applying these concepts creatively can lead to innovative solutions across numerous disciplines.

Integral Calculus

Integral Calculus deals with concepts and techniques for integrating functions, which essentially involves computing areas, volumes, and summations. The integral of a function over an interval gives the accumulated area under the curve represented by the function.

In the context of integral calculus, numerical and approximate methods, like the trapezoidal rule, provide practical approaches to handle complex or unsolvable integrals analytically. Integral calculus forms the backbone of advanced calculus studies, linking directly to differential calculus. It plays a crucial role in solving differential equations and understanding the behavior of dynamic systems. Thus, mastering integral calculus, including its numerical methods, is instrumental in achieving mathematical proficiency.