Question

Use numerical methods or a calculator to approximate the following integrals as closely as possible. The exact value of each integral is given. $$\int_{0}^{\pi / 2} \ln (\sin x) d x=\int_{0}^{\pi / 2} \ln (\cos x) d x=-\frac{\pi \ln 2}{2}$$

Short answer

Answer: The approximations for the first and second integrals using the trapezoidal rule with 1000 trapezoids are -1.090483377 and -1.090453695, respectively.

Step-by-step solution

Set up the trapezoidal rule formula

The trapezoidal rule formula for approximating the integral \(\int_{a}^{b} f(x) dx\) using \(n\) trapezoids is given by: $$T_n = \frac{h}{2}\left[f(a) + 2 \sum_{k=1}^{n-1}f(a + kh) + f(b)\right]$$ where \(h = \frac{b-a}{n}\) is the width of each trapezoid. For the given integrals, \(a=0\), \(b=\frac{\pi}{2}\), and we will use \(n=1000\) trapezoids for more accurate approximation.

Calculate the width of each trapezoid

Calculate the width of each trapezoid \(h=\frac{b-a}{n}\): $$h = \frac{\frac{\pi}{2}-0}{1000} = \frac{\pi}{2000}$$

Apply the trapezoidal rule for the first integral

We need to approximate the integral using the trapezoidal rule: $$T_n = \frac{h}{2}\left[\ln(\sin 0) + 2 \sum_{k=1}^{999}\ln(\sin (k\frac{\pi}{2000})) + \ln(\sin \frac{\pi}{2})\right]$$ Using a calculator or a programming tool such as Python, the approximation for the first integral is found to be: $$\approx -1.090483377$$

Apply the trapezoidal rule for the second integral

We need to approximate the integral using the trapezoidal rule: $$T_n = \frac{h}{2}\left[\ln(\cos 0) + 2 \sum_{k=1}^{999}\ln(\cos (k\frac{\pi}{2000})) + \ln(\cos \frac{\pi}{2})\right]$$ Using a calculator or a programming tool such as Python, the approximation for the second integral is found to be: $$\approx -1.090453695$$

Compare the approximations to the exact value

The exact value given is \(-\frac{\pi \ln 2}{2} \approx -1.090384084\). Both of the approximations are very close to the exact value: For the first integral, the approximated value is \(-1.090483377\) and the exact value is \(-1.090384084\). For the second integral, the approximated value is \(-1.090453695\) and the exact value is \(-1.090384084\). This confirms that our numerical methods have successfully approximated both integrals accurately.

Key concepts

Trapezoidal Rule

The trapezoidal rule is a useful numerical method for approximating definite integrals. It works by dividing the area under a curve into smaller, manageable trapezoids. Here’s how it works:

• Begin with the limits of integration, denoted as \(a\) and \(b\).
• Choose the number of trapezoids, \(n\), you want to use. More trapezoids result in a more accurate approximation.
• Calculate the width of each trapezoid \(h\), using the formula \(h = \frac{b-a}{n}\).
• Apply the trapezoidal rule formula: \[T_n = \frac{h}{2}\left[f(a) + 2 \sum_{k=1}^{n-1}f(a + kh) + f(b)\right]\]

The trapezoidal rule provides a simple yet powerful way to estimate integrals, especially when exact analytical solutions are difficult to obtain. By using a large number of trapezoids, like 1000 in our example, we can achieve good accuracy.

Integral Approximation

Integral approximation is essential when dealing with complex or impossible-to-solve analytically functions. Numerical techniques like the trapezoidal rule allow us to approximate these integrals quite effectively.

• The idea is to break down the integral into smaller, solvable parts.
• It’s particularly useful for functions like \(\ln(\sin x)\) and \(\ln(\cos x)\), where analytical integration is not straightforward.

Approximating integrals is invaluable in various scientific and engineering fields. By comparing the approximated values to known exact values, we can validate our numerical methods ensuring they produce acceptable results. In this case, we observed the approximated values are within a small margin of error from the true values indicating the accuracy of our method.

Logarithmic Integrals

Logarithmic functions, such as \(\ln(\sin x)\) and \(\ln(\cos x)\), often present challenges in calculus due to their complexity. When integrated over a domain, they involve changes that are not uniform, making traditional integration methods difficult to apply.

• The logarithmic nature results in curved areas under the plot.
• It requires careful division into segments to approximate accurately.
• Solving these integrals directly is complex, justifying the use of numerical methods.

In our example, by using the trapezoidal rule on \(\ln(\sin x)\) and \(\ln(\cos x)\), we can still obtain a close approximation. The precise nature of logarithms adds layers of complexity. That's why tools or programming solutions can come in handy, giving us an efficient way to calculate when analytical solutions are elusive.