Question

Approximate the following integrals using either the midpoint rule, trapezoidal rule, or Simpson's rule as indicated. (Round answers to three decimal places.) \(\int_{1}^{2} \frac{d x}{x} ;\) trapezoidal rule; \(n=5\)

Short answer

The approximate value is 0.696.

Step-by-step solution

Identify the Method and Parameters

We need to approximate the integral using the trapezoidal rule with 5 (=5) intervals. The integral is \( \int_{1}^{2} \frac{1}{x} \,dx \).

Calculate the Interval Width

To apply the trapezoidal rule, first calculate the interval width \( h \). This is given by \( h = \frac{b-a}{n} \), where \( a = 1 \), \( b = 2 \), and \( n = 5 \). Thus, \( h = \frac{2-1}{5} = 0.2 \).

Determine the Sample Points

Next, determine the sample points where we will evaluate the function. They are given by \( x_0 = 1, \ x_1 = 1.2, \ x_2 = 1.4, \ x_3 = 1.6, \ x_4 = 1.8, \) and \( x_5 = 2 \).

Evaluate the Function at Sample Points

Evaluate the function \( f(x) = \frac{1}{x} \) at each sample point:- \( f(x_0) = \frac{1}{1} = 1 \)- \( f(x_1) = \frac{1}{1.2} \approx 0.833 \)- \( f(x_2) = \frac{1}{1.4} \approx 0.714 \)- \( f(x_3) = \frac{1}{1.6} \approx 0.625 \)- \( f(x_4) = \frac{1}{1.8} \approx 0.556 \)- \( f(x_5) = \frac{1}{2} = 0.5 \) (rounded to three decimal places).

Apply the Trapezoidal Rule

The trapezoidal rule is applied using:\[ T = \frac{h}{2} \left( f(x_0) + 2\left( f(x_1) + f(x_2) + f(x_3) + f(x_4) \right) + f(x_5) \right) \]Substituting the values, we get:\[ T = \frac{0.2}{2} \left( 1 + 2(0.833 + 0.714 + 0.625 + 0.556) + 0.5 \right) \]

Compute the Result

Calculate the result of the expression:\[ T = 0.1 \left( 1 + 2(2.728) + 0.5 \right) \]\[ T = 0.1 \left( 1 + 5.456 + 0.5 \right) \]\[ T = 0.1 \times 6.956 \]\[ T = 0.6956 \]Rounded to three decimal places, the approximated integral value is 0.696.

Key concepts

Trapezoidal Rule Simplified

The trapezoidal rule is a numerical method used to approximate the value of definite integrals, especially when finding an exact integral is difficult. It approximates the area under a curve by dividing it into trapezoids instead of using the exact calculus integral technique. Each trapezoid is formed by two linear segments connecting consecutive points on the curve.

The steps for using the trapezoidal rule include:

• First, identify the limits of integration: the interval \([a, b]\)
• Divide the interval into small sections or intervals, with each having width \(h = \frac{b-a}{n}\), where \('n'\) is the number of intervals
• Calculate the function values at each endpoint
• Apply the trapezoidal rule formula, \[T = \frac{h}{2} \left( f(x_0) + 2\left( f(x_1) + f(x_2) + \cdots + f(x_{n-1}) \right) + f(x_n) \right)\]

Calculating the areas of these trapezoids provides an approximation to the integral's value. This method is efficient and simple, making it practical for many real-world applications where accuracy can be balanced with computational simplicity.

The trapezoidal rule is typically more accurate than simpler methods like the midpoint rule for a given number of divisions, providing a balance between ease of implementation and result precision.

Understanding Approximation Methods

Approximation methods, such as the trapezoidal rule, play a crucial role in solving integrals that are otherwise difficult to evaluate analytically. These techniques are essential in fields like engineering, physics, and computational sciences, where precise measurements are necessary but exact solutions may not be feasible for complex equations.

• These methods create estimates by breaking complex functions into simpler, manageable pieces.
• Key approximation methods include the midpoint rule, trapezoidal rule, and Simpson's rule, each offering specific benefits depending on the problem's requirements.
• Trapezoidal rule uses linear approximations between points, leading to a straightforward application, whereas methods like Simpson's rule may use parabolic approximations for potentially greater accuracy.

Using approximation methods introduces some error into calculations, known as "truncation error." However, increasing the number of intervals used in the approximation can reduce this error, improving accuracy.

The balance between simplicity and accuracy often determines the choice of method. For example, if computational resources are limited or a quick estimation is needed, the trapezoidal rule may suffice. In contrast, for higher precision, more detailed methods might be used.

What are Integrals?

Integrals are fundamental to calculus, representing accumulation of quantities and the area under curves. The integral of a function from point \(a\) to \(b\) gives the net area between the function graph and the x-axis over the interval \( [a, b] \).

There are two main types of integrals:

• Definite integrals: These have upper and lower limits, evaluating the net area under a curve within these limits. They are noted as \(\int_a^b f(x)\, dx\). This is the type used with the trapezoidal rule, as in the provided exercise.
• Indefinite integrals: Represent the family of all antiderivatives of a function, not having specific bounds. They are denoted by \(\int f(x)\, dx = F(x) + C\), where \(C\) is the constant of integration.

The process of integration is closely related to differentiation, essentially being the reverse or inverse operation. Where differentiation breaks down functions into rates of change, integration builds up to determine accumulations.

Practical applications of integrals are vast, from computing areas in physics to determining volumes and solving equations in engineering. In essence, integrals help in visualizing and calculating dimensions in both theoretical and applied sciences.