Question

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

Short answer

The approximate integral is 1.566.

Step-by-step solution

Understand Simpson's Rule

Simpson's Rule is a numerical method to approximate the value of a definite integral. The formula is: \[\int_{a}^{b} f(x) \, dx \approx \frac{b-a}{3n} \left(f(x_0) + 4 \sum_{i=1, \text{ odd}}^{n-1} f(x_i) + 2 \sum_{i=2, \text{ even}}^{n-2} f(x_i) + f(x_n)\right)\]where \( n \) must be even, \( x_0 = a \), \( x_n = b \), and \( x_i = a + i\cdot h \) where \( h = \frac{b-a}{n} \).

Determine Parameters

Given \( a = 0.1 \), \( b = 0.5 \), and \( n = 4 \). Calculate \( h = \frac{b-a}{n} = \frac{0.5-0.1}{4} = 0.1 \).This means the interval is divided into 4 segments of length 0.1.

Calculate Function Values

Determine the values of the function \( f(x) = \frac{\cos x}{x} \) at the points: \( x_0 = 0.1 \)\( x_1 = 0.2 \)\( x_2 = 0.3 \)\( x_3 = 0.4 \)\( x_4 = 0.5 \). Calculate:- \( f(x_0) = \frac{\cos(0.1)}{0.1} \approx 9.950 \)- \( f(x_1) = \frac{\cos(0.2)}{0.2} \approx 4.875 \)- \( f(x_2) = \frac{\cos(0.3)}{0.3} \approx 3.281 \)- \( f(x_3) = \frac{\cos(0.4)}{0.4} \approx 2.302 \)- \( f(x_4) = \frac{\cos(0.5)}{0.5} \approx 1.755 \).

Apply Simpson's Rule Formula

Substitute the function values into Simpson's Rule formula:\[\int_{0.1}^{0.5} \frac{\cos x}{x} \, dx \approx \frac{0.4}{12} \left(9.950 + 4 \cdot (4.875 + 2.302) + 2 \cdot 3.281 + 1.755\right)\]Calculate:- 4 times the sum of odd terms: \(4 \cdot (4.875 + 2.302) = 28.708\)- 2 times the sum of even terms: \(2 \cdot 3.281 = 6.562\)Then add these to the first and last function values: \( 9.950 + 28.708 + 6.562 + 1.755 = 46.975 \).

Final Calculation

Now, multiply by \( \frac{0.4}{12} \): \[ \frac{0.4}{12} \cdot 46.975 \approx 1.566 \]Thus, the approximate value of the integral is 1.566, rounded to three decimal places.

Key concepts

Numerical Integration

Numerical integration is a fundamental concept in calculus and numerical analysis. It involves finding the approximate value of a definite integral, which represents the accumulation of quantities like area under a curve. In cases where the exact integral is difficult or impossible to compute analytically, numerical integration provides a practical solution.

To approach this, one utilizes various methods or rules that compute an estimate of the area by summing up simple geometric shapes like rectangles or trapezoids. One such popular method is Simpson's Rule, which uses parabolic arcs instead of straight lines for better accuracy.

Numerical integration is crucial in fields where the integral form of a function is unknown or complex. It's also vital in real-world applications across physics, engineering, and economics, where it allows for solving problems involving rates of change, probabilities, and accumulated growth.

Definite Integrals

A definite integral is a fundamental tool in calculus used to calculate the net area under a curve, or the total accumulation of a quantity over an interval. When dealing with a continuous function within a closed interval \(a, b\), the definite integral adds up the infinitesimal sums of the curve.

The notation for definite integrals is \( \int_a^b f(x) \, dx\), where \(f(x)\) is the function being integrated, and \(a\) and \(b\) are the limits of integration. The result is a single numerical value that represents the net sum of the function's values across the specified interval.

Understanding the concept of definite integrals is essential, as it provides the mathematical foundation for solving problems involving total distance traveled, total energy consumption, or any other cumulative measure. By using numerical methods like Simpson's Rule, one can approximate these integrals even when they cannot be evaluated explicitly.

Approximation Methods

Approximation methods in calculus are techniques used to find nearly precise numerical solutions to problems that are challenging to solve exactly. They are especially useful for integrals with no elementary antiderivative or in situations involving complex functions. There are several types of approximation methods, each with a specific approach and varying degrees of accuracy.

• Trapezoidal Rule: This method approximates the area under a curve by dividing it into trapezoids rather than rectangles, providing a more accurate estimate than a simple Riemann sum.
• Simpson's Rule: It uses parabolic segments to estimate the area under a curve, resulting in higher accuracy for smooth functions. Simpson's Rule is particularly effective when the function being integrated is well-behaved and smooth over the interval.
• Midpoint Rule: This method estimates the integral by taking the function's value at the midpoint of each subinterval.

Approximation methods are indispensable when one needs to evaluate integrals in practical situations where the function's form is intricate or the exact evaluation is computationally intensive. Using these techniques helps in obtaining reliable solutions in a range of scientific and engineering problems.