Question

[T] Use a calculator to approximate \(\int_{0}^{1} \sin (\pi x) d x\) using the midpoint rule with 25 subdivisions. Compute the relative error of approximation.

Short answer

The approximation is about 0.6366, with a relative error of approximately 0.0086.

Step-by-step solution

Understand the Midpoint Rule

The midpoint rule is a technique to approximate the integral of a function. We divide the interval into equal subintervals. Then, instead of taking the endpoints, we use the midpoint of each subinterval to calculate the function value. We sum these values and multiply by the width of the subintervals to estimate the integral.

Set Up the Subintervals

Divide the interval \([0, 1]\) into 25 subintervals. Each subinterval will have a width \(\Delta x = \frac{1 - 0}{25} = 0.04\).

Determine Midpoints

The midpoint for each subinterval \([x_{i-1}, x_i]\) with width \(\Delta x\) is given by \(x_i^* = x_{i-1} + \frac{\Delta x}{2}\). For the first subinterval, this is \(x_1^* = 0 + 0.02 = 0.02\), and similarly for each subsequent subinterval, it increases by \(\Delta x = 0.04\). Thus, \(x_i^* = 0.02 + (i-1)\times 0.04\).

Calculate Function Values at Midpoints

Evaluate \( \sin(\pi x_i^*) \) for each midpoint. These values will be used in the next step. For example, \( \sin(\pi \times 0.02) \), \( \sin(\pi \times 0.06) \), etc., up to the 25th midpoint.

Apply the Midpoint Rule Formula

The midpoint rule formula is \( \int_{a}^{b} f(x) \, dx \approx \Delta x \sum_{i=1}^{n} f(x_i^*) \). Here, \(\Delta x = 0.04\), and you substitute each calculated value from Step 4 into the formula: \[\int_{0}^{1} \sin(\pi x)dx \approx 0.04 \left( \sin(\pi \times 0.02) + \sin(\pi \times 0.06) + \ldots + \sin(\pi \times 0.98) \right)\].

Compute the Exact Integral

Find the exact value using calculus: \[\int_{0}^{1} \sin(\pi x) \, dx = -\frac{1}{\pi} \cos(\pi x) \bigg|_{0}^{1} = -\frac{1}{\pi} [\cos(\pi \times 1) - \cos(\pi \times 0)] = \frac{2}{\pi}\].

Calculate the Relative Error

With the approximate value from Step 5 and the exact value from Step 6, compute the relative error using the formula: \[ \text{Relative Error} = \frac{|\text{Exact Value} - \text{Approximate Value}|}{|\text{Exact Value}|} \]. Substitute the values to find the relative error.

Key concepts

Numerical Integration

Numerical integration is a mathematical process used to find approximate solutions to integrals. When functions are complex or difficult to integrate analytically, numerical methods like the midpoint rule come to the rescue. The midpoint rule is an example of a Riemann sum method, which involves approximating the area under a curve by dividing the interval of integration into several smaller, equally spaced subintervals.

In the midpoint rule, each subinterval is represented by a rectangle. Instead of using the endpoint values, you take the function value at the midpoint of each subinterval. This midvalue gives a better estimate of the function's behavior over the subinterval. The process involves:

• Dividing the interval into a set number of subintervals.
• Calculating the midpoints of each subinterval.
• Evaluating the function at each midpoint.
• Multiplying each function value by the width of the subinterval to compute the area of each rectangle.
• Summing up these areas to approximate the total integral.

This technique is particularly useful in cases where exact integration is infeasible. It provides a balance between accuracy and computational efficiency.

Relative Error

When we approximate integrals, it is crucial to understand the accuracy of our solution. This is where relative error comes into play. Relative error helps us quantify how close our approximation is to the actual value. It is an important measure in numerical computations, providing insight into the reliability of the result.

Relative error is calculated by taking the absolute difference between the exact value and the approximate value, then dividing by the absolute exact value. The formula is:\[ \text{Relative Error} = \frac{|\text{Exact Value} - \text{Approximate Value}|}{|\text{Exact Value}|} \]This fraction gives us a dimensionless number, usually expressed as a percentage, indicating the magnitude of the error relative to the size of the exact value. A smaller relative error means a more accurate approximation.

When refining the approximation of an integral, you may decrease the step size or increase the number of subdivisions to reduce the relative error, thus achieving greater precision.

Trigonometric Integration

Trigonometric functions are often used in mathematics and physics, and integrating these functions can sometimes be challenging. Trigonometric integration involves finding the integral of functions including sine, cosine, tangent, and other related functions.

In this case, the integration of a sine function over a specific interval is evaluated. The exact solution for the integral of \( \sin\), particularly in this exercise, can be found manually using mathematical techniques like antiderivatives. The sine function's antiderivative is the negative cosine function, which allows us to find the exact solution with definite limits:
\[ \int \sin(\pi x) \, dx = -\frac{1}{\pi} \cos(\pi x) + C \]where \(C\) is the constant of integration. For definite integrals like \( \int_{0}^{1} \sin(\pi x) \, dx \), the constant \(C\) is not needed. By evaluating the antiderivative at the upper and lower limits, you can find the exact value of the integral without approximation.

Understanding trigonometric integration is fundamental when working with waves, oscillations, and periodic phenomena, making it a vital skill in both theoretical and applied contexts.