Problem 6 - Single Variable Calculus

Question

Use (a) the Trapezoidal Rule and (b) Simpson's Rule to approximate the integral. Compare your results with the exact value of the integral. $$ \int_{0}^{1} e^{-x} d x ; \quad n=6 $$

Step-by-Step Solution

Answer

We find the interval width, \(\Delta x = \frac{1}{6}\), and approximate the integral \(\int_{0}^{1} e^{-x} dx\) using the Trapezoidal Rule and Simpson's Rule: a) Trapezoidal Rule: \(\int_{0}^{1} e^{-x} dx \approx 0.631102\) b) Simpson's Rule: \(\int_{0}^{1} e^{-x} dx \approx 0.632120\) The exact value of the integral is approximately \(0.632121\). Comparing the approximations, we can see that Simpson's Rule provides a closer approximation to the exact value than the Trapezoidal Rule in this case.

Step 1: Calculate interval width (Δx)

Divide the interval [0, 1] into six equal parts. The interval width is given by: $$ \Delta x = \frac{(b - a)}{n} = \frac{1-0}{6} = \frac{1}{6} $$

Step 2: Use the Trapezoidal Rule

Apply the Trapezoidal Rule formula: $$ \int_{a}^{b} f(x) dx \approx \frac{\Delta x}{2} \left[f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n)\right] $$ Compute the function values: $$ f(x) = e^{-x} $$ Then, approximate the integral using the Trapezoidal Rule: \[ \begin{aligned} \int_{0}^{1} e^{-x} dx &\approx \frac{1}{12} \left[e^{-0} + 2(e^{-\frac{1}{6}} + e^{-\frac{2}{6}} + e^{-\frac{3}{6}} + e^{-\frac{4}{6}} + e^{-\frac{5}{6}}) + e^{-1}\right] \\ &\approx 0.631102 \end{aligned} \]

Step 3: Use Simpson's Rule

Apply Simpson's Rule formula: $$ \int_{a}^{b} f(x) dx \approx \frac{\Delta x}{3} \left[f(x_0) + 4\sum_{i=1}^{n/2} f(x_{2i-1}) + 2\sum_{i=1}^{n/2-1} f(x_{2i}) + f(x_n)\right] $$ Then, approximate the integral using Simpson's Rule: \[ \begin{aligned} \int_{0}^{1} e^{-x} dx &\approx \frac{1}{18} \left[e^{-0} + 4(e^{-\frac{1}{6}} + e^{-\frac{3}{6}} + e^{-\frac{5}{6}}) + 2(e^{-\frac{2}{6}} + e^{-\frac{4}{6}}) + e^{-1}\right] \\ &\approx 0.632120 \end{aligned} \]

Step 4: Find the exact value of the integral

Integrate the function exactly: $$ \int_{0}^{1} e^{-x} dx = \left[-e^{-x}\right]_{0}^{1} = -(e^{-1} - e^0) = 1 - e^{-1} $$ The exact value of the integral is approximately 0.632121.

Step 5: Compare the results

Comparing the approximations with the exact value: - Trapezoidal Rule approximation: 0.631102 - Simpson's Rule approximation: 0.632120 - Exact value of the integral: 0.632121 Simpson's Rule provides a closer approximation to the exact value of the integral than the Trapezoidal Rule in this case.

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a numerical method used to estimate the definite integral of a function. When you don't have the luxury of solving an integral analytically, this method comes in handy. Essentially, the Trapezoidal Rule approximates the region under the curve by dividing it into a series of trapezoids, hence the name.

• Each trapezoid is formed by using the function values at the endpoints of the subintervals as the heights.
• The formula used for approximation is: \[ \int_{a}^{b} f(x) dx \approx \frac{\Delta x}{2} \left[f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n)\right] \]
• To apply this method, you need to calculate the width of each subinterval, \(\Delta x\).

In the exercise above, with \(n=6\) subintervals spanning from 0 to 1, the interval width \(\Delta x\) is \(\frac{1}{6}\). Substituting the function \(f(x) = e^{-x}\) and using the formula, the approximate value was computed to be 0.631102. While the method is simple and generally provides a rough estimate, its accuracy improves with a larger number of trapezoids. It offers a good compromise between simplicity and precision, but it may not be as precise as some more sophisticated methods.

Simpson's Rule

Simpson's Rule is another powerful numerical technique to approximate definite integrals. It is often more accurate than the Trapezoidal Rule due to its use of parabolic arcs instead of straight lines to approximate the function's curve.

• Simpson's Rule assumes that over each pair of subintervals, the graph of the function can be fit by a parabola.
• The formula for Simpson’s Rule is given by: \[ \int_{a}^{b} f(x) dx \approx \frac{\Delta x}{3} \left[f(x_0) + 4\sum_{i=1}^{n/2} f(x_{2i-1}) + 2\sum_{i=1}^{n/2-1} f(x_{2i}) + f(x_n)\right] \]

Applying this to the same function \(f(x) = e^{-x}\) with \(\Delta x = \frac{1}{6}\) and \(n=6\), Simpson's Rule yields an approximation of 0.632120. The precision of Simpson's Rule typically surpasses that of the Trapezoidal Rule for most well-behaved functions when the number of intervals is moderately large. That's because it takes into account more information about the function's curvature by considering additional points within the interval.

Definite Integration

Definite integration seeks to compute the exact area under the curve of a given function over a specified interval. This is represented mathematically by the integral symbol with lower and upper bounds, \([a, b]\), indicating the start and end points of the interval.

• The purpose of definite integration is to determine the accumulation of quantities, such as area, volume, and other quantifiable aspects.
• In the context of our problem, we are working to integrate \(\int_{0}^{1} e^{-x} dx\).

Exact integration is often achievable with analytical methods when the integral is straightforward, like the exponential function involved here. Here, the exact value was computed to be \(1 - e^{-1}\), which is approximately 0.632121. However, for more complex functions, or when a quick approximation is required, numerical methods like the Trapezoidal and Simpson's Rule are employed. These methods become particularly valuable when it’s challenging to find a symbolic integral.

Approximation Methods

Approximation methods are central in scenarios where calculating the exact value of an integral is either inconvenient or impossible. These techniques bridge the gap between calculus' theoretical world and their practical applications in science and engineering.

• They provide a pragmatic means to estimate values, particularly useful when dealing with empirical data where functions may not be explicitly known.
• Common methods include the Trapezoidal Rule and Simpson’s Rule—each employed based on the requirements for speed, simplicity, or accuracy.
• These methods work by discretizing the interval and making intelligent guesses about the shape of the integrand between known points.

In our exercise, both rules provided estimates of an exponential function's integral over \([0, 1]\). Simpson's Rule was more precise, demonstrating how certain methods can outperform others depending on the function's nature and the interval size. These methods play a crucial role in computational fields and are essential tools for engineers, scientists, and financial analysts alike, making complex integral calculations feasible and interpretable.