Evaluating Integral ∫(0 To 1)√(1-x³) Dx Using Trapezoidal Rule

Introduction to Numerical Integration

In many areas of mathematics, physics, and engineering, evaluating definite integrals is a fundamental task. While some integrals can be solved analytically using techniques like substitution, integration by parts, or trigonometric identities, many integrals, especially those involving complex functions or lacking elementary antiderivatives, require numerical methods for their evaluation. One such numerical method is the Trapezoidal rule, a simple yet powerful technique for approximating the definite integral of a function. This article delves into the application of the Trapezoidal rule to evaluate the integral ${ \int_0^1 \sqrt{1 - x^3} \, dx }$, taking six equal intervals and aiming for accuracy up to two decimal places. This integral is particularly interesting because the function ${ \sqrt{1 - x^3} }$ does not have a simple elementary antiderivative, making numerical methods like the Trapezoidal rule essential for its evaluation. Understanding how to apply these numerical techniques is crucial for solving real-world problems where analytical solutions are not feasible.

The Trapezoidal rule approximates the area under a curve by dividing the interval of integration into smaller trapezoids and summing their areas. This method is based on approximating the function within each subinterval by a straight line, which forms the top of the trapezoid. The more trapezoids we use (i.e., the smaller the width of each interval), the closer the approximation will be to the true value of the integral. This article provides a detailed step-by-step guide on how to apply this rule, highlighting the importance of choosing an appropriate number of intervals to achieve the desired accuracy. We will discuss the theoretical underpinnings of the Trapezoidal rule, its advantages and limitations, and how it compares to other numerical integration methods. The specific integral ${ \int_0^1 \sqrt{1 - x^3} \, dx }$ serves as an excellent example to illustrate the practical application of the Trapezoidal rule, showcasing its effectiveness in handling integrals that are challenging to solve analytically.

The process of evaluating the integral ${ \int_0^1 \sqrt{1 - x^3} \, dx }$ using the Trapezoidal rule involves several key steps. First, we need to divide the interval of integration [0, 1] into six equal subintervals. This determines the width of each trapezoid. Next, we calculate the function values at the endpoints of each subinterval. These values represent the heights of the trapezoids. Then, we apply the Trapezoidal rule formula, which involves summing the function values at the endpoints, with the interior points counted twice, and multiplying by half the width of the subintervals. This calculation gives us an approximation of the definite integral. Finally, we ensure that our result is accurate up to two decimal places, which may involve refining the number of subintervals or using error estimation techniques. Throughout this article, we will emphasize the importance of careful computation and attention to detail to achieve accurate results. We will also discuss the sources of error in numerical integration and how to minimize them, providing a comprehensive understanding of the Trapezoidal rule and its application in practical scenarios.

The Trapezoidal Rule

The Trapezoidal rule is a numerical integration technique used to approximate the definite integral of a function. It works by dividing the area under the curve of the function into a series of trapezoids and summing their areas. This method provides a straightforward way to estimate the value of an integral when an analytical solution is difficult or impossible to find. The accuracy of the Trapezoidal rule depends on the number of trapezoids used; more trapezoids generally lead to a more accurate approximation. This section will delve into the mathematical formulation of the Trapezoidal rule, its underlying principles, and the factors that influence its accuracy. Understanding the theoretical basis of the Trapezoidal rule is essential for its effective application and for interpreting the results obtained. The rule is widely used in various fields, including engineering, physics, and computer science, to solve problems involving integration. Its simplicity and ease of implementation make it a valuable tool for numerical analysis.

The fundamental idea behind the Trapezoidal rule is to approximate the function ${ f(x) }$ over each subinterval by a linear function. Geometrically, this means replacing the curve of the function with a straight line segment. The area under this line segment forms a trapezoid, and the sum of the areas of these trapezoids provides an approximation of the definite integral. The formula for the Trapezoidal rule is derived by considering the area of each trapezoid, which is given by the average of the function values at the endpoints of the subinterval multiplied by the width of the subinterval. When we sum the areas of all the trapezoids, we obtain the composite Trapezoidal rule formula. This formula is a weighted sum of the function values at the endpoints and interior points of the interval of integration. The weights are chosen such that the formula accurately approximates the integral. The accuracy of the approximation increases as the width of the subintervals decreases, which corresponds to using more trapezoids. This relationship between the number of trapezoids and the accuracy of the approximation is a key aspect of the Trapezoidal rule.

The mathematical formulation of the Trapezoidal rule can be expressed as follows: Let ${ I = \int_a^b f(x) \, dx }$ be the definite integral we want to approximate. Divide the interval [a, b] into n equal subintervals, each of width ${ h = \frac{b - a}{n} }$. Let ${ x_i = a + ih }$ for ${ i = 0, 1, ..., n }$ be the endpoints of the subintervals. The Trapezoidal rule approximation of the integral I is then given by: ${ I \approx \frac{h}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)] }$ This formula shows that the function values at the interior points ${ x_1, x_2, ..., x_{n-1} }$ are weighted twice, while the function values at the endpoints ${ x_0 }$ and ${ x_n }$ are weighted once. This weighting scheme reflects the fact that the interior points are shared by two trapezoids, while the endpoints are only part of one trapezoid. The factor of ${ \frac{h}{2} }$ comes from the average height of the trapezoids. Understanding this formula is crucial for applying the Trapezoidal rule correctly and for interpreting the results obtained. The formula provides a clear and concise way to approximate definite integrals using a numerical method that is both simple and effective.

Applying the Trapezoidal Rule to ${ \int_0^1 \sqrt{1 - x^3} \, dx }$

To evaluate the integral ${ \int_0^1 \sqrt{1 - x^3} \, dx }$ using the Trapezoidal rule with six equal intervals, we first need to divide the interval [0, 1] into six subintervals. This involves calculating the width of each subinterval and the endpoints of these subintervals. Then, we evaluate the function ${ f(x) = \sqrt{1 - x^3} }$ at these endpoints. These function values will be used in the Trapezoidal rule formula to approximate the integral. This section provides a detailed step-by-step guide on how to perform these calculations and apply the formula. We will emphasize the importance of careful computation to ensure the accuracy of the result. The application of the Trapezoidal rule to this specific integral serves as a practical example of how to use this numerical method to solve problems that are difficult or impossible to solve analytically.

First, we divide the interval [0, 1] into six equal subintervals. The width of each subinterval, denoted by h, is given by: ${ h = \frac{b - a}{n} = \frac{1 - 0}{6} = \frac{1}{6} }$ where a = 0, b = 1, and n = 6. Next, we determine the endpoints of the subintervals, which are given by ${ x_i = a + ih }$ for ${ i = 0, 1, 2, 3, 4, 5, 6 }$. These endpoints are: ${ \begin{aligned} x_0 &= 0 + 0(\frac{1}{6}) = 0 \\ x_1 &= 0 + 1(\frac{1}{6}) = \frac{1}{6} \\ x_2 &= 0 + 2(\frac{1}{6}) = \frac{2}{6} = \frac{1}{3} \\ x_3 &= 0 + 3(\frac{1}{6}) = \frac{3}{6} = \frac{1}{2} \\ x_4 &= 0 + 4(\frac{1}{6}) = \frac{4}{6} = \frac{2}{3} \\ x_5 &= 0 + 5(\frac{1}{6}) = \frac{5}{6} \\ x_6 &= 0 + 6(\frac{1}{6}) = 1 \end{aligned} }$ Now, we evaluate the function ${ f(x) = \sqrt{1 - x^3} }$ at each of these endpoints. These function values are crucial for applying the Trapezoidal rule formula.

Next, we calculate the values of the function ${ f(x) = \sqrt{1 - x^3} }$ at the endpoints ${ x_i }$ we calculated earlier. These values are necessary for applying the Trapezoidal rule formula. We have: ${ \begin{aligned} f(x_0) &= f(0) = \sqrt{1 - 0^3} = \sqrt{1} = 1 \\ f(x_1) &= f(\frac{1}{6}) = \sqrt{1 - (\frac{1}{6})^3} = \sqrt{1 - \frac{1}{216}} = \sqrt{\frac{215}{216}} \approx 0.997683 \\ f(x_2) &= f(\frac{1}{3}) = \sqrt{1 - (\frac{1}{3})^3} = \sqrt{1 - \frac{1}{27}} = \sqrt{\frac{26}{27}} \approx 0.981456 \\ f(x_3) &= f(\frac{1}{2}) = \sqrt{1 - (\frac{1}{2})^3} = \sqrt{1 - \frac{1}{8}} = \sqrt{\frac{7}{8}} \approx 0.935414 \\ f(x_4) &= f(\frac{2}{3}) = \sqrt{1 - (\frac{2}{3})^3} = \sqrt{1 - \frac{8}{27}} = \sqrt{\frac{19}{27}} \approx 0.838900 \\ f(x_5) &= f(\frac{5}{6}) = \sqrt{1 - (\frac{5}{6})^3} = \sqrt{1 - \frac{125}{216}} = \sqrt{\frac{91}{216}} \approx 0.649523 \\ f(x_6) &= f(1) = \sqrt{1 - 1^3} = \sqrt{1 - 1} = 0 \end{aligned} }$ These function values are essential for the next step, where we will apply the Trapezoidal rule formula to approximate the integral. Accurate calculation of these values is crucial for obtaining a reliable approximation of the integral.

Calculation using the Trapezoidal Rule Formula

Now that we have calculated the function values at the endpoints of the subintervals, we can apply the Trapezoidal rule formula to approximate the integral ${ \int_0^1 \sqrt{1 - x^3} \, dx }$. The Trapezoidal rule formula, as discussed earlier, involves a weighted sum of the function values, multiplied by half the width of the subintervals. This section provides a step-by-step calculation using the formula and the function values we obtained. We will substitute the values into the formula and perform the arithmetic operations to arrive at the approximation. The accuracy of the approximation will depend on the number of subintervals used, which in this case is six. Careful application of the formula and accurate calculations are essential for obtaining a reliable estimate of the integral.

The Trapezoidal rule formula is given by: ${ I \approx \frac{h}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + 2f(x_4) + 2f(x_5) + f(x_6)] }$ where ${ h = \frac{1}{6} }$ is the width of the subintervals, and ${ f(x_i) }$ are the function values we calculated earlier. Substituting the values into the formula, we get: ${ \begin{aligned} I &\approx \frac{1/6}{2} [1 + 2(0.997683) + 2(0.981456) + 2(0.935414) + 2(0.838900) + 2(0.649523) + 0] \\ &\approx \frac{1}{12} [1 + 1.995366 + 1.962912 + 1.870828 + 1.677800 + 1.299046 + 0] \\ &\approx \frac{1}{12} [9.805952] \\ &\approx 0.817163 \end{aligned} }$ Therefore, the approximation of the integral ${ \int_0^1 \sqrt{1 - x^3} \, dx }$ using the Trapezoidal rule with six equal intervals is approximately 0.817163. This result provides an estimate of the area under the curve of the function ${ f(x) = \sqrt{1 - x^3} }$ between x = 0 and x = 1. The accuracy of this approximation can be further improved by increasing the number of subintervals, which would reduce the width of each trapezoid and provide a closer fit to the curve.

Final Result and Accuracy

Based on the calculations using the Trapezoidal rule with six equal intervals, we have approximated the integral ${ \int_0^1 \sqrt{1 - x^3} \, dx }$ to be approximately 0.817163. However, the question requires the answer to be correct up to two decimal places. Therefore, we need to round our result to two decimal places. This section focuses on presenting the final result and discussing the accuracy of the approximation. We will consider the potential sources of error in the Trapezoidal rule and how they affect the final answer. Additionally, we will briefly discuss methods for improving the accuracy of the approximation, such as increasing the number of subintervals or using other numerical integration techniques.

Rounding the result 0.817163 to two decimal places, we get 0.82. Therefore, the approximate value of the integral ${ \int_0^1 \sqrt{1 - x^3} \, dx }$ using the Trapezoidal rule with six equal intervals, correct up to two decimal places, is 0.82. This is our final answer.

The accuracy of the Trapezoidal rule approximation depends on several factors, including the number of subintervals used and the nature of the function being integrated. The Trapezoidal rule is a second-order method, meaning that its error is proportional to the square of the width of the subintervals. In general, increasing the number of subintervals will improve the accuracy of the approximation, but it will also increase the computational effort. For the integral ${ \int_0^1 \sqrt{1 - x^3} \, dx }$, using six subintervals provides a reasonable approximation, but it is possible to obtain a more accurate result by using a larger number of subintervals or by employing a higher-order numerical integration method, such as Simpson's rule. Simpson's rule approximates the function using quadratic polynomials instead of linear functions, which generally leads to a more accurate approximation for smooth functions.

In conclusion, we have successfully evaluated the integral ${ \int_0^1 \sqrt{1 - x^3} \, dx }$ using the Trapezoidal rule with six equal intervals. The approximate value of the integral, correct up to two decimal places, is 0.82. While this result provides a good estimate of the integral, it is important to recognize the limitations of the Trapezoidal rule and the potential for error. For applications requiring higher accuracy, it may be necessary to use more sophisticated numerical integration techniques or to increase the number of subintervals. The Trapezoidal rule, however, remains a valuable tool for approximating integrals, especially when analytical solutions are not available.