Question

Approximate the integral to three decimal places using the indicated rule. \(\int_{0}^{0.4} \sin \left(x^{2}\right) d x ;\) trapezoidal rule; \(n=4\)

Short answer

The approximate integral is 0.044.

Step-by-step solution

Determine the Interval Width

The interval over which we are integrating is from 0 to 0.4. With n=4 subintervals, we calculate the width of each subinterval (h) using the formula: \( h = \frac{b-a}{n} \) where \( b = 0.4 \) and \( a = 0 \). Thus, \( h = \frac{0.4 - 0}{4} = 0.1 \).

Identify the Subinterval Points

For \( n = 4 \) and \( h = 0.1 \), the points are: \( x_0 = 0 \), \( x_1 = 0.1 \), \( x_2 = 0.2 \), \( x_3 = 0.3 \), and \( x_4 = 0.4 \). These are the points where we will evaluate the function \( \sin(x^2) \).

Evaluate the Function at Each Point

Calculate the value of the function \( \sin(x^2) \) at each subinterval point: - \( f(x_0) = \sin(0^2) = \sin(0) = 0 \)- \( f(x_1) = \sin((0.1)^2) = \sin(0.01) \)- \( f(x_2) = \sin((0.2)^2) = \sin(0.04) \)- \( f(x_3) = \sin((0.3)^2) = \sin(0.09) \)- \( f(x_4) = \sin((0.4)^2) = \sin(0.16) \)

Apply the Trapezoidal Rule Formula

The trapezoidal rule formula is given by:\[\int_{a}^{b} f(x) \, dx \approx \frac{h}{2} \left[ f(x_0) + 2(f(x_1) + f(x_2) + f(x_3)) + f(x_4) \right]\]Substitute the evaluated function values and \( h = 0.1 \) into the formula:\[\frac{0.1}{2} \left[ 0 + 2(\sin(0.01) + \sin(0.04) + \sin(0.09)) + \sin(0.16) \right]\]

Calculate the Approximate Integral

Using a calculator, find the approximate values for the sine functions: \( \sin(0.01) \approx 0.01 \), \( \sin(0.04) \approx 0.04 \), \( \sin(0.09) \approx 0.09 \), \( \sin(0.16) \approx 0.159 \). Sum these values following the trapezoidal rule formula:\[\frac{0.1}{2} [0 + 2(0.01 + 0.04 + 0.09) + 0.159] \]Simplify:\[= \frac{0.1}{2} [0 + 2(0.14) + 0.159] \]\[= \frac{0.1}{2} [0.28 + 0.159] \]\[= \frac{0.1}{2} \times 0.439 \]\[= 0.0439 \]Thus, the approximate value of the integral is 0.044.

Key concepts

Numerical Integration

Numerical integration is a method used in calculus to calculate the value of an integral when its exact solution is difficult to find analytically. This approach is particularly useful for functions that do not have elementary antiderivatives or when dealing with real-world data that doesn't conform to a straightforward mathematical formula.
In essence, numerical integration approximates the area under a curve over a specified interval. Various techniques exist to achieve this approximation, each with its own balance between simplicity and accuracy. The **Trapezoidal Rule** is one such method that provides a good trade-off between computational efficiency and reasonably accurate results.
Numerical integration becomes essential in scenarios requiring precision, such as in engineering and physical sciences when measuring quantities like distance, area, or volume from complex shapes or curves. Understanding how these numerical methods work helps in choosing the most suitable approach, depending on the specific needs of a problem.

Approximate Integral Calculation

Approximate integral calculation is central to numerical integration, offering solutions where analytical calculations are impractical. The goal is to find a value that is close enough to the true integral for all practical purposes, allowing computations to proceed with the confidence that errors are minimal and controlled.
One widely used method is the **Trapezoidal Rule**. This rule works by approximating the region under a curve as a series of adjacent trapezoids, calculating their areas, and summing these areas to form an estimate of the total integral.
Here’s a quick rundown of how the process works:

• Divide the interval \( [a, b] \) into \( n \) small subintervals.
• Approximate the curve by straight lines connecting the points on the function over these subintervals.
• Calculate the area under the straight lines.
• Sum these areas to get the approximate integral.

The approximation improves with more subintervals but at the cost of increasing computation time.

Calculus Methods

Calculus methods, like the Trapezoidal Rule, serve as fundamental tools for integrating complex functions. These methods provide a way to numerically approximate integrals, catering to scenarios where functions are not easily handled by symbolic computation.
The **Trapezoidal Rule** is merely one of several methods of numerical integration available. Some other popular methods include:

• **Simpson's Rule**, which generally offers more accurate results by fitting parabolas to segments of the curve rather than straight lines.
• **Midpoint Rule**, which uses midpoint heights of subintervals to estimate the integral.
• **Gaussian Quadrature**, which provides high accuracy for a broad class of functions but is more complex to implement.

Choosing a method depends on factors like the desired accuracy, the nature of the function, and the computational resources available. By understanding these different calculus methods, students and professionals can select the appropriate strategy for various problems, ensuring efficient and accurate computations.