Problem 14
Question
In Exercises \(11-14\) , use the tabulated values of the integrand to estimate the integral with (a) the Trapezoidal Rule and (b) Simpson's Rule with \(n=8\) steps. Round your answers to five decimal places. Then (c) find the integral's exact value and the approximation error \(E_{T}\) or \(E_{s}\) as appropriate. $$ \int_{\pi / 4}^{\pi / 2}\left(\csc ^{2} y\right) \sqrt{\cot y} d y $$
Step-by-Step Solution
Verified Answer
Approximate the integral with Trapezoidal and Simpson's Rule using given formulas next; find exact computational value and errors.
1Step 1: Understanding the Problem
We need to estimate the integral \( \int_{\pi / 4}^{\pi / 2} \left( \csc^{2} y \right) \sqrt{\cot y} \, dy \) using (a) the Trapezoidal Rule and (b) Simpson's Rule with \(n=8\) intervals. Then (c) compute the exact value and the approximation errors for both methods.
2Step 2: Define the Functions
The integrand is \( f(y) = \left( \csc^{2} y \right) \sqrt{\cot y} \). We need to evaluate this function at specific points to use the numerical methods. Calculate \( f(y) \) at each node for \( n=8 \) intervals. The range is \( y \) from \( \pi/4 \) to \( \pi/2 \).
3Step 3: Calculate the Step Size
The interval \((\pi/4, \pi/2)\) is split into \(n=8\) equal parts. The step size \( h \) is calculated as: \[ h = \frac{\pi/2 - \pi/4}{8} = \frac{\pi}{16}. \]
4Step 4: Apply the Trapezoidal Rule
The formula for the Trapezoidal Rule is: \[ T = \frac{h}{2} \left( f(y_0) + 2 \sum_{i=1}^{n-1} f(y_i) + f(y_n) \right). \] Calculate \( f(y) \) at each interval point and apply the formula to estimate the integral.
5Step 5: Apply Simpson's Rule
The formula for Simpson's Rule is: \[ S = \frac{h}{3} \left( f(y_0) + 4 \sum_{i=1, \, \text{odd}}^{n-1} f(y_i) + 2 \sum_{i=2, \, \text{even}}^{n-2} f(y_i) + f(y_n) \right). \] Use the values of \( f(y) \) calculated earlier, along with the step size, to estimate the integral using Simpson's Rule.
6Step 6: Compute the Exact Integral
Calculate the exact value of the integral \( \int_{\pi/4}^{\pi/2} \left( \csc^{2} y \right) \sqrt{\cot y} \, dy \) using analytical techniques or a symbolic computation tool.
7Step 7: Estimate the Approximation Errors
The approximation errors are \( E_T = |I - T| \) for the Trapezoidal Rule and \( E_S = |I - S| \) for Simpson's Rule, where \( I \) is the exact value of the integral. Calculate both errors using the values obtained in the previous steps.
Key Concepts
Trapezoidal RuleSimpson's RuleApproximation ErrorIntegrand Estimation
Trapezoidal Rule
The Trapezoidal Rule is one of the simplest numerical methods used for approximating the definite integral of a function. It works by approximating the region under the graph of the function as a series of trapezoids and then summing their areas to estimate the total integral. This method provides a straightforward way to approach integrals when finding the exact analytical solution is difficult.
In our exercise, we approximate the integral of \( \int_{\pi / 4}^{\pi / 2} \left( \csc^{2} y \right) \sqrt{\cot y} \, dy \) using 8 intervals between \( \pi/4 \) and \( \pi/2 \). The formula used for this method is:
\[T = \frac{h}{2} \left( f(y_0) + 2 \sum_{i=1}^{n-1} f(y_i) + f(y_n) \right),\]
where \( h \) is the step size, which divides our interval into equal parts. One of the advantages of the Trapezoidal Rule is its simplicity and the improvement of accuracy with increased intervals. However, it's worth noting that it may not be as precise for functions with high variability in the integration interval.
In our exercise, we approximate the integral of \( \int_{\pi / 4}^{\pi / 2} \left( \csc^{2} y \right) \sqrt{\cot y} \, dy \) using 8 intervals between \( \pi/4 \) and \( \pi/2 \). The formula used for this method is:
\[T = \frac{h}{2} \left( f(y_0) + 2 \sum_{i=1}^{n-1} f(y_i) + f(y_n) \right),\]
where \( h \) is the step size, which divides our interval into equal parts. One of the advantages of the Trapezoidal Rule is its simplicity and the improvement of accuracy with increased intervals. However, it's worth noting that it may not be as precise for functions with high variability in the integration interval.
Simpson's Rule
Simpson's Rule is another popular method for numerical integration, known for its improved accuracy over the Trapezoidal Rule. This approach involves approximating segments of the integrand by second-degree polynomials (parabolas) instead of straight lines, which can better fit the curvature of the function.
In our particular example, we use Simpson's Rule with \( n=8 \) intervals to estimate the integral of \( \int_{\pi / 4}^{\pi / 2} \left( \csc^{2} y \right) \sqrt{\cot y} \, dy \). The formula for Simpson's Rule is:
\[ S = \frac{h}{3} \left( f(y_0) + 4 \sum_{i=1, \, \text{odd}}^{n-1} f(y_i) + 2 \sum_{i=2, \, \text{even}}^{n-2} f(y_i) + f(y_n) \right). \]
This method is typically more accurate because it captures function curvature through quadratic approximations. It is especially effective when the function is smooth within the integration bounds, making it a preferred choice for many practical applications.
In our particular example, we use Simpson's Rule with \( n=8 \) intervals to estimate the integral of \( \int_{\pi / 4}^{\pi / 2} \left( \csc^{2} y \right) \sqrt{\cot y} \, dy \). The formula for Simpson's Rule is:
\[ S = \frac{h}{3} \left( f(y_0) + 4 \sum_{i=1, \, \text{odd}}^{n-1} f(y_i) + 2 \sum_{i=2, \, \text{even}}^{n-2} f(y_i) + f(y_n) \right). \]
This method is typically more accurate because it captures function curvature through quadratic approximations. It is especially effective when the function is smooth within the integration bounds, making it a preferred choice for many practical applications.
Approximation Error
When using numerical methods like the Trapezoidal and Simpson's Rules, it's crucial to evaluate the accuracy of these approximations. The approximation error measures the difference between the exact integral value and the estimated value from our numerical calculations.
For our exercise, the approximation errors are determined by:
Understanding approximation errors allows us to assess how close our numerical methods are to the true value. Generally, smaller errors indicate higher accuracy, making this evaluation essential for deciding which numerical method best suits a particular problem.
For our exercise, the approximation errors are determined by:
- \( E_T = |I - T| \) for the Trapezoidal Rule
- \( E_S = |I - S| \) for Simpson's Rule
Understanding approximation errors allows us to assess how close our numerical methods are to the true value. Generally, smaller errors indicate higher accuracy, making this evaluation essential for deciding which numerical method best suits a particular problem.
Integrand Estimation
Integrand estimation plays a vital role in numerical integration, providing the values needed at specific points of the function to apply methods like the Trapezoidal and Simpson's Rules. These estimates are determined at chosen points, determined by the partitioning of the integral's interval.
For the exercise at hand, the integrand is given by \( f(y) = \left( \csc^{2} y \right) \sqrt{\cot y} \). These function values are calculated at every nodal point within our defined interval from \( \pi/4 \) to \( \pi/2 \), with \( n=8 \) as determined by the step size \( h = \frac{\pi}{16} \).
Accurate estimation of the integrand ensures that the subsequent numerical integration methods can be performed reliably, providing more precise results. Each estimated function value serves as an important part of the interpolation used in the numerical method.
For the exercise at hand, the integrand is given by \( f(y) = \left( \csc^{2} y \right) \sqrt{\cot y} \). These function values are calculated at every nodal point within our defined interval from \( \pi/4 \) to \( \pi/2 \), with \( n=8 \) as determined by the step size \( h = \frac{\pi}{16} \).
Accurate estimation of the integrand ensures that the subsequent numerical integration methods can be performed reliably, providing more precise results. Each estimated function value serves as an important part of the interpolation used in the numerical method.
Other exercises in this chapter
Problem 13
Evaluate each integral in Exercises \(1-36\) by using a substitution to reduce it to standard form. $$ \int \sec \frac{t}{3} d t $$
View solution Problem 14
Evaluate the integrals in Exercises \(1-28\). $$ \int \frac{2 d x}{x^{3} \sqrt{x^{2}-1}}, \quad x>1 $$
View solution Problem 14
Use the table of integrals at the back of the book to evaluate the integrals. \(\int \frac{\sqrt{x^{2}-4}}{x} d x\)
View solution Problem 14
Evaluate the integrals in Exercises \(1-34\) without using tables. $$ \int_{-\infty}^{\infty} \frac{x d x}{\left(x^{2}+4\right)^{3 / 2}} $$
View solution