Problem 3 - Calculus: Early Transcendental Functions

Question

Compute Midpoint, Trapezoidal and Simpson's Rule approximations by hand (leave your answer as a fraction) for \(n=4\). $$\int_{1}^{3} \frac{1}{x} d x$$

Step-by-Step Solution

Answer

Midpoint Rule approximation is \( \frac{828}{1250}\), Trapezoidal Rule approximation is \(\frac{23}{30}\), and Simpson's Rule approximation is \( \frac{1447}{1800}\).

Step 1: Subdivide the Interval

The given interval [1, 3] is divided into \(n=4\) equal subintervals, which results in the points 1, 1.5, 2, 2.5, 3.

Step 2: Midpoint Rule

The Midpoint Rule formula is \(h \sum_{i=1}^{n} f \left( \frac{x_{i-1}+x_{i}}{2} \right)\). Here, \(h=\frac{b-a}{n}=\frac{3-1}{4}=0.5\). Therefore, the approximation is \(0.5 \left( \frac{1}{1.25} + \frac{1}{1.75} + \frac{1}{2.25} + \frac{1}{2.75} \right) = \frac{828}{1250}\).

Step 3: Trapezoidal Rule

The Trapezoidal Rule formula is \(h \left( \frac{f(a)+f(b)}{2} + \sum_{i=1}^{n-1} f(x_{i}) \right)\). Hence, the approximation is \(0.5 \left( \frac{1/1 + 1/3}{2} + \frac{1}{1.5} + \frac{1}{2} + \frac{1}{2.5} \right) = \frac{23}{30}\).

Step 4: Simpson's Rule

Simpson's Rule formula is \(\frac{h}{3} \left( f(a) + 4 \sum_{i=1, odd}^{n} f(x_{i}) + 2 \sum_{i=2, even}^{n-1} f(x_{i}) + f(b) \right)\). The approximation is \(\frac{0.5}{3} \left( 1/1 + 4 \left( \frac{1}{1.5} + \frac{1}{2.5} \right) + 2 \left( \frac{1}{2} \right) + 1/3 \right) = \frac{1447}{1800}\).

Key Concepts

Midpoint Rule

The Midpoint Rule is one of the easiest ways to approximate the integral of a function. It uses the midpoint of each subinterval to estimate the area under the curve. To apply this rule to our exercise, we start by dividing the interval \( [1, 3] \) into 4 equal parts, resulting in subintervals. The width of each subinterval is denoted as \( h \), calculated as \( \frac{b-a}{n} \).
In this example, \( h = \frac{3 - 1}{4} = 0.5 \). We then identify the midpoints of these subintervals: 1.25, 1.75, 2.25, and 2.75. Using the function \( f(x) = \frac{1}{x} \), we calculate the function value at these midpoints. Finally, we sum up these values and multiply by \( h \) to get our approximation:

• \( h \cdot \Big( f(1.25) + f(1.75) + f(2.25) + f(2.75) \Big) \)
• Approximation: \( 0.5 \left( \frac{1}{1.25} + \frac{1}{1.75} + \frac{1}{2.25} + \frac{1}{2.75} \right) = \frac{828}{1250} \)

By using the Midpoint Rule, we approximate the integral quickly and with reasonable accuracy.

Trapezoidal Rule

The Trapezoidal Rule provides a more accurate approximation than the Midpoint Rule because it uses linear segments to approximate the area under the curve. It divides the total area into trapezoids, rather than rectangles, and then sums the areas of these trapezoids. In this exercise, we apply the same subintervals and calculate the height \( h \) as \( 0.5 \), similar to the Midpoint Rule.
To begin, we calculate the function values at the endpoints and interior points:

• \( f(1), f(3) \)
• The midpoints of subinterval are \( f(1.5), f(2), f(2.5) \)

Then, apply the Trapezoidal Rule: \[ h \left( \frac{f(1) + f(3)}{2} + f(1.5) + f(2) + f(2.5) \right) \]
For the integral \( \int_{1}^{3} \frac{1}{x} \, dx \), plug the function values into the formula:\[ 0.5 \left( \frac{1/1 + 1/3}{2} + 1/1.5 + 1/2 + 1/2.5 \right) = \frac{23}{30} \] This method effectively averages out the steepness of the curve, providing a better approximation.

Simpson's Rule

Simpson's Rule is a more sophisticated technique that often gives the most accurate approximation among the three methods. By utilizing parabolas to estimate the area under the curve, it provides a closer approximation for functions that display curvature.
For the interval \([1, 3]\), divided into 4 parts, we calculate with \( h = 0.5 \). Simpson's Rule formula is: \[ \frac{h}{3} \left( f(a) + 4 \sum_{i=1, \text{odd}}^{n} f(x_{i}) + 2 \sum_{i=2, \text{even}}^{n-1} f(x_{i}) + f(b) \right) \]

• Function values are computed at the interval points: \( f(1), f(1.5), f(2), f(2.5), f(3) \)

Plug in these values:\[ \frac{0.5}{3} \Big( 1/1 + 4(1/1.5 + 1/2.5) + 2(1/2) + 1/3 \Big) = \frac{1447}{1800} \] Simpson's Rule provides a remarkably accurate approximation for the given integral, as it efficiently handles both linear and non-linear features of the function.