Problem 12 - Calculus for Biology and Medicine

Question

Use the trapezoidal rule to approximate each integral with the specified value of \(n .\) \(\int_{0}^{\pi / 2} \sin x d x, n=4\)

Step-by-Step Solution

Answer

The approximate value of the integral is 0.9978.

Step 1: Understand the Trapezoidal Rule

The trapezoidal rule approximates the integral by dividing the area under the curve into trapezoids. If we have an interval from \(a\) to \(b\) divided into \(n\) subintervals, the trapezoidal rule states: \[ \int_{a}^{b} f(x) \, dx \approx \frac{b-a}{2n} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right] \] where \(x_0, x_1, \ldots, x_n\) are the values at the endpoints and each subinterval division.

Step 2: Set up for the Problem

Here, the function is \(\sin x\), the interval is from \(0\) to \(\frac{\pi}{2}\), and \(n = 4\). The width of each subinterval (\(h\)) is given by \(h = \frac{\frac{\pi}{2} - 0}{4} = \frac{\pi}{8}\). The points are: \(x_0 = 0\), \(x_1 = \frac{\pi}{8}\), \(x_2 = \frac{\pi}{4}\), \(x_3 = \frac{3\pi}{8}\), and \(x_4 = \frac{\pi}{2}\).

Step 3: Calculate Function Values

Now, calculate \(\sin x\) at each point: - \(f(x_0) = \sin(0) = 0\)- \(f(x_1) = \sin(\frac{\pi}{8})\)- \(f(x_2) = \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\)- \(f(x_3) = \sin(\frac{3\pi}{8})\)- \(f(x_4) = \sin(\frac{\pi}{2}) = 1\)

Step 4: Apply the Trapezoidal Rule

Plug the values into the trapezoidal rule formula: \[ \int_{0}^{\frac{\pi}{2}} \sin x \, dx \approx \frac{\frac{\pi}{2}}{8} \left[ 0 + 2 \left( \sin\left(\frac{\pi}{8}\right) + \frac{\sqrt{2}}{2} + \sin\left(\frac{3\pi}{8}\right) \right) + 1 \right]. \]Evaluate and sum the sine values to approximate the integral.

Step 5: Calculate Final Value

After calculating, \(\int_{0}^{\frac{\pi}{2}} \sin x \, dx \approx \frac{\pi}{16} \left[ 1 + 2 \left( \sin\left(\frac{\pi}{8}\right) + \frac{\sqrt{2}}{2} + \sin\left(\frac{3\pi}{8}\right) \right) \right] \).This simplifies the expression and calculates to roughly 0.9978.

Key Concepts

Numerical Integration

Numerical integration refers to a wide variety of algorithms used to approximate the definite integral of a function, especially when an exact analytical solution is difficult or impossible to obtain.
It is often employed when functions are complex or data is in discrete form. This technique has many useful applications in science and engineering where exact mathematical forms may not be immediately solvable.

• The main idea is to estimate the area under the curve by discretizing it into simpler geometric shapes (like a trapezoid in the case of the trapezoidal rule).
• Numerical integration can involve techniques like the trapezoidal rule, Simpson's rule, and others, each having its own level of precision and complexity.
• Choosing the right numerical integration method may depend on the function's behavior and the required accuracy.

This makes numerical integration a powerful alternative to traditional calculus, especially in handling real-world data that might not fit neat mathematical expressions.

Approximation Methods

The trapezoidal rule is a classic example of an approximation method within calculus that helps find the integral of a function.
This method involves breaking down a complex integral into simpler parts, making it manageable to solve. The trapezoidal rule achieves this by approximating the area under a curve using trapezoids rather than perfect curves.

• Approximation methods are not always exact, but they can be made more accurate by increasing the number of partitions ( ).
• The trade-off is that more calculations are required with an increase in partitions, potentially requiring computational resources.
• Different approximation methods, besides the trapezoidal rule, include Midpoint Rule and Simpson’s Rule, each improving upon accuracy by different means.

Understanding various approximation methods equipped students with the tools for tackling many calculus problems that would otherwise be too complex to handle directly.

Calculus

Calculus forms the backbone of numerical methods and approximation techniques, with its fundamental concepts being applied in these contexts.
The study of calculus deals with changes and motion and can transform complex real-world phenomena into understandable equations.

• Integrals are a core component of calculus, representing the accumulation of quantities, such as areas, volumes, and other limits of sums.
• The trapezoidal rule, used in the given exercise, leverages basic calculus to produce a simple method for approximating integrals.
• Understanding calculus is essential not only for solving mathematical problems but also for applying these solutions in physics, engineering, and economics critically.

This integration of calculus into numerical methods allows us to address widespread practical problems through innovative and adaptable means.