Problem 1 - Thomas Calculus

Question

The instructions for the integrals in Exercises \(1-10\) have two parts, one for the Trapezoidal Rule and one for Simpson's Rule. I. Using the Trapezoidal Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{T}\right| .\) b. Evaluate the integral directly and find \(\left|E_{T}\right|\) c. Use the formula \((\left | E_{T}\right | /(\) true value)) \(\times 100\) to express \(\left|E_{T}\right|\) as \right.\right. a percentage of the integral's true value. II. Using Simpson's Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{S}\right| .\) b. Evaluate the integral directly and find \(\left|E_{S}\right|\) c. Use the formula \((\left|E_{S}\right| /(\) true value) \() \times 100\) to express \(\left|E_{S}\right|\) as \right. \(\quad\) a percentage of the integral's true value. $$ \int_{1}^{2} x d x $$

Step-by-Step Solution

Answer

Both methods have approximations of 1.5 with errors of 0%.

Step 1: Define the Function and the Integral

We are required to estimate the integral \( \int_{1}^{2} x \, dx \) using both the Trapezoidal Rule and Simpson's Rule. For both methods, we will then calculate the errors and express them as percentages of the true value of the integral.

Step 2: True Value of the Integral

Evaluate the integral directly to find its true value. \[\int_{1}^{2} x \, dx = \left[ \frac{x^2}{2} \right]_{1}^{2} = \frac{4}{2} - \frac{1}{2} = 1.5\]The true value of the integral is 1.5.

Step 3: Trapezoidal Rule Approximation

The Trapezoidal Rule for estimating the integral \( \int_{a}^{b} f(x) \, dx \) with \( n \) steps is given by:\[T_n = \frac{b-a}{2n} \left(f(a) + 2\sum_{i=1}^{n-1} f(x_i) + f(b) \right)\]For \( n=4 \) and \( f(x) = x \), \( a=1 \), \( b=2 \):\( h = (b-a)/n = 0.25 \)\( x_0=1, \, x_1=1.25, \, x_2=1.5, \, x_3=1.75, \, x_4=2 \)\[T_4 = \frac{0.25}{2} (1 + 2(1.25 + 1.5 + 1.75) + 2) = \frac{0.25}{2} (1 + 2(4.5) + 2) = 1.5\]The estimated value using the Trapezoidal Rule is 1.5.

Step 4: Error Calculation for Trapezoidal Rule

The error for the Trapezoidal Rule \( |E_T| \) is given by:\[|E_T| \leq \frac{(b-a)^3}{12n^2} M_2\]where \( M_2 \) is the maximum absolute value of the second derivative of \( f(x) \) over \([a, b]\). Since \( f(x) = x \), \( f''(x) = 0 \), it follows that \( M_2 = 0 \).Therefore, \(|E_T| = 0\).The actual error \(|E_T|\) when compared to the true value is zero (as calculated in the previous step).

Step 5: Percentage Error for Trapezoidal Rule

The percentage error is\[\left ( \frac{|E_T|}{\text{true value}} \right ) \times 100 = \left( \frac{0}{1.5} \right) \times 100 = 0\%\]

Step 6: Simpson's Rule Approximation

Simpson's Rule for estimating the integral \( \int_{a}^{b} f(x) \, dx \) with \( n \) even steps is given by:\[S_n = \frac{b-a}{3n} \left(f(a) + 4\sum_{i=1, \text{odd}}^{n-1} f(x_i) + 2\sum_{i=2, \text{even}}^{n-2} f(x_i) + f(b) \right)\]Using \( n=4 \):\[S_4 = \frac{1}{12} (1 + 4(1.25 + 1.75) + 2(1.5) + 2) = \frac{1}{12} (1 + 4(3) + 3 + 2) = 1.5\]The estimated value using Simpson's Rule is also 1.5.

Step 7: Error Calculation for Simpson's Rule

The error for Simpson's Rule \( |E_S| \) is given by:\[|E_S| \leq \frac{(b-a)^5}{180n^4} M_4\] \( M_4 \) is the maximum absolute value of the fourth derivative of \( f(x) \) over \([a, b]\). For \( f(x) = x \), all derivatives are zero after the first derivative, hence \( M_4 = 0 \).Thus, \(|E_S| = 0\).The actual error \(|E_S|\) when compared to the true value is also zero.

Step 8: Percentage Error for Simpson's Rule

The percentage error is\[\left ( \frac{|E_S|}{\text{true value}} \right ) \times 100 = \left( \frac{0}{1.5} \right) \times 100 = 0\%\]

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a straightforward numerical integration method used to approximate the definite integral of a function. This method involves dividing the area under the curve into trapezoids rather than rectangles or other shapes. Here’s how it works:

• Divide the integration interval \([a, b]\) into equal smaller subintervals.
• Apply the function to the endpoints of each subinterval.
• Use the average of these values to calculate the area of each trapezoid.

For the integral \(\int_{1}^{2} x \, dx\), with \((n = 4)\) steps:

• The width of each subinterval (h) is calculated as \(h = (b-a)/n = 0.25\).
• The function is evaluated at \(x_0, x_1, x_2, x_3, x_4\) which are \(1, 1.25, 1.5, 1.75, 2\).

The estimated integral is calculated as \(T_4 = 1.5\). This method is particularly useful for functions that are not easily integrated analytically.

Simpson's Rule

Simpson's Rule is another popular method for numerical integration, which typically provides more accurate results than the Trapezoidal Rule. It uses parabolic arcs instead of straight line segments to approximate the function:

• Divide the interval \([a, b]\) into an even number of subintervals of equal width.
• Apply the function values at these points.
• Fit parabolas through sequences of three points.

For the integral \(\int_{1}^{2} x \, dx\) with \(n=4\), the estimated integral utilizing Simpson's Rule is also 1.5. This method, \[S_4 = \frac{1}{12} (1 + 4(1.25 + 1.75) + 2(1.5) + 2) = 1.5\], offers improved accuracy, especially for smooth functions because it better captures the curvature of a function.

Error Estimation

Error estimation is crucial for assessing the accuracy of numerical integration methods. Both the Trapezoidal and Simpson's Rules have specific formulas to estimate the potential error:

• Trapezoidal Rule Error: \[|E_T| \leq \frac{(b-a)^3}{12n^2} M_2\] where \(M_2\) is the maximum of the second derivative of \(f(x)\).
• Here, since \(f''(x) = 0\), \(M_2\) is zero, leading to zero error.
• Simpson's Rule Error: \[|E_S| \leq \frac{(b-a)^5}{180n^4} M_4\] where \(M_4\) is the maximum of the fourth derivative of \(f(x)\).
• Again, with higher derivatives of \(f(x) = x\) being zero, the error is zero.

In both cases, error estimation determines that the numerical approximations match the true integral value perceive.

Definite Integral

A definite integral represents the area under a curve within a specific interval \([a, b]\). It gives the accumulated total of a quantity over this range and is vital in applications such as physics, engineering and finance.
For example, given \(\int_{1}^{2} x \, dx = 1.5\), the definite integral calculates the exact area under the graph of \(f(x) = x\) from \(x = 1\) to \(x = 2\). Calculating this directly involves finding the antiderivative and evaluating it at the bounds:

• Antiderivative function: \[\frac{x^2}{2}\]
• Substitute and solve: \[\left[ \frac{x^2}{2} \right]_{1}^{2} = \frac{4}{2} - \frac{1}{2} = 1.5\]

This provides the exact integral value needed for comparison to numerical methods.

Approximation Methods

Approximation methods like the Trapezoidal and Simpson's Rules form the foundation of numerical integration.

• These methods provide practical alternatives to analytical integration, especially for complex functions or those without standard antiderivatives.
• While they do estimate the integral value, the accuracy can often be controlled by adjusting the number of subintervals \(n\).
• The trade-off includes increased computational work with more intervals.

For students and professionals alike, understanding these approximation techniques is essential for tackling real-world problems involving curve integration when analytical methods are impractical.