Question

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

Short answer

The approximate integral is 0.021.

Step-by-step solution

Understand the Problem

We need to approximate the integral \( \int_{0}^{0.4} \sin(x^2) \, dx \) using Simpson's Rule with \( n = 4 \) subintervals. Simpson's Rule involves the formula: \[ \int_{a}^{b} f(x) \, dx \approx \frac{b-a}{3n} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + \ldots + 4f(x_{n-1}) + f(x_n) \right] \].

Calculate the Subinterval Width, h

First, calculate the width \( h \) of each subinterval. Given \( a = 0 \), \( b = 0.4 \), and \( n = 4 \), use the formula \( h = \frac{b-a}{n} \). This gives \( h = \frac{0.4 - 0}{4} = 0.1 \).

Determine the x-values

Calculate the x-values at which the function will be evaluated: \( x_0 = 0, x_1 = 0.1, x_2 = 0.2, x_3 = 0.3, \) and \( x_4 = 0.4 \) by adding \( h = 0.1 \) consecutively to the starting point \( a = 0 \) until \( b = 0.4 \).

Evaluate the Function

Evaluate the function \( \sin(x^2) \) at all these points:- \( f(x_0) = \sin(0^2) = \sin(0) = 0 \)- \( f(x_1) = \sin(0.1^2) = \sin(0.01) \approx 0.01 \)- \( f(x_2) = \sin(0.2^2) = \sin(0.04) \approx 0.04 \)- \( f(x_3) = \sin(0.3^2) = \sin(0.09) \approx 0.09 \)- \( f(x_4) = \sin(0.4^2) = \sin(0.16) \approx 0.159 \).

Apply Simpson's Rule Formula

Substitute the function values into Simpson's Rule Formula:\[ \int_{0}^{0.4} \sin(x^2) \, dx \approx \frac{0.4-0}{3 \times 4} \left[ 0 + 4(0.01) + 2(0.04) + 4(0.09) + 0.159 \right] \].

Perform the Calculations

Compute the expression:\( = \frac{0.4}{12} [0 + 0.04 + 0.08 + 0.36 + 0.159] = \frac{0.4}{12} \times 0.639 \).Calculate the result: \( \frac{0.4}{12} \approx 0.0333 \) and \( 0.0333 \times 0.639 \approx 0.0213 \).

Present the Final Answer

The approximate value of the integral \( \int_{0}^{0.4} \sin(x^2) \, dx \) using Simpson's Rule with \( n = 4 \) is approximately 0.021 to three decimal places.

Key concepts

Numerical Integration

Numerical integration is a key technique used to approximate the value of an integral when the exact form is difficult or impossible to find analytically. This method is especially useful for integrals involving complex functions or those defined over specific intervals without returning to elementary functions. When performing numerical integration, we aim to compute an approximate area under the curve.

Naturally, there are many approaches to numerical integration, each with varying degrees of complexity and precision. Some are simple and fast, suited for rough approximations, while others are more complex, offering greater precision. Whether you're estimating the path of a spacecraft or calculating financial derivatives, numerical integration provides the means to tackle integrals effectively.

Approximation Methods

In the world of numerical integration, approximation methods are the cornerstone of practical calculations. These methods allow us to estimate the value of difficult integrals by breaking them into easier, solvable parts.

• **Trapezoidal Rule:** It approximates the area under the curve as a series of trapezoids, rather straightforward and easy to understand, though less accurate over large intervals.
• **Simpson's Rule:** Utilizes parabolic arcs rather than straight lines for approximation, offering improved accuracy. Simpson's Rule is effective when the function is smooth and can be subdivided suitably into even numbers of intervals.

Both methods provide a way to handle complex functions by bringing them into simpler, manageable segments. With Simpson's Rule, the method is particularly efficient because it uses a weighted sum of the function's values that assumes a quadratic form, making it better suited for smooth, oscillating functions.

Subinterval Calculation

When using approximation methods such as Simpson's Rule, computing subintervals is a critical step. The particular interval, \(\[a, b\]\), is divided into smaller sections called subintervals. Each interval captures a portion of the function's behavior, contributing to the overall approximation.

To determine the width of each subinterval, we apply the formula \(h = \frac{b-a}{n} \), where \(n \) is the number of subintervals. For instance, in our example, with \(a = 0 \), \(b = 0.4\), and \(n = 4\), this yields \(h = 0.1\). With these increments, the necessary function values can be determined at regular points \(x_0, x_1, x_2, x_3,\) and \(x_4\). Calculating the subintervals ensures the function's behavior is accurately captured across its range, leading to better approximation outcomes for the entire integral.

Trapezoidal Rule Alternative

Simpson’s Rule serves as a great alternative to the Trapezoidal Rule, especially when dealing with functions that can be nicely approximated by parabolas. While the Trapezoidal Rule is simpler and uses linear approximations, Simpson’s Rule takes it a step further by leveraging quadratic fitting. This extra step improves the precision significantly.

The essence of Simpson's Rule lies in fitting parabolas through segments of the function to estimate the area more accurately. This offers a significant advantage when dealing with curves that change direction, such as sinusoidal functions, where linear approximation might miss intricate details. The technique of alternating between multipliers of 4 and 2 across subintervals contributes to the efficiency and accuracy, making it superior in contexts requiring more refined analysis over evenly spaced intervals.