Problem 19 - College Algebra and Calculus: An Applied Approach

Question

Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of \(n\). (Round your answers to three significant digits.) $$ \int_{0}^{1} \sqrt{1-x^{2}} d x, n=4 $$

Step-by-Step Solution

Answer

a) The trapezoidal rule approximation is 0.788. b) The Simpson's rule approximation is 0.785.

Step 1: Calculate Trapezoidal Rule Approximation

Firstly, compute the width of each trapezoid, \( \Delta x = (b - a) / n \) where \( a = 0 \), \( b = 1 \), and \( n = 4 \). After computing \( \Delta x \), apply the trapezoidal rule: \[ T_n = \Delta x [0.5f(a) + 0.5f(b) + f(x_1) + f(x_2) + ... + f(x_{n-1})] \] where \( f(x) = \sqrt{1 - x^2} \] and \( x_i = a + i * \Delta x \). Calculate \( f(x_i) \) for \( i \in [1,3] \) and substitute into the equation, remembering to apply rounding as instructed.

Step 2: Calculate Simpson's Rule Approximation

Secondly, now calculate the width of each parabola segment, which is same as previous \( \Delta x \). Then apply Simpson's rule: \[ S_n = \Delta x [f(a) + 4*f(x_1) + 2*f(x_2) + 4*f(x_3) + f(b)]/3 \] where \( f(x) = \sqrt{1 - x^2} \] and \( x_i = a + i * \Delta x \). Calculate \( f(x_i) \) for \( i \in [1,3] \) and substitute into the equation, remembering to apply rounding as instructed.

Step 3: Round and State the Final Results

With both the trapezoidal and Simpson's rule approximations calculated, round each to three decimal places and provide them as the final answer to the problem.

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a numerical method used to estimate the value of a definite integral. It is particularly useful when the function is difficult to integrate analytically. The fundamental idea is to replace the area under the curve with a series of trapezoids, making it easier to compute.

In our exercise, the problem involves finding an approximation for the definite integral \( \int_{0}^{1} \sqrt{1-x^{2}} \, dx \) with \( n = 4 \). To use the Trapezoidal Rule, we start by dividing the interval \([0, 1]\) into 4 equal parts. The width of each trapezoid is \( \Delta x = \frac{1-0}{4} = 0.25 \).

The next step is to evaluate the function \( f(x) = \sqrt{1-x^2} \) at each of these points: \( x_0 = 0 \), \( x_1 = 0.25 \), \( x_2 = 0.5 \), \( x_3 = 0.75 \), and \( x_4 = 1 \). Evaluate and sum the function values at these points in the weighted manner as per the trapezoidal formula:

• \( T_n = \Delta x [0.5f(x_0) + f(x_1) + f(x_2) + f(x_3) + 0.5f(x_4)] \)

This gives us an estimated integral, rounded to the required significant digits.

Simpson's Rule

Simpson's Rule is another technique for numeric integration that provides an even more accurate approximation than the Trapezoidal Rule. This method assumes the function behaves like a series of parabolic arcs rather than straight lines. It is suitable for situations where the function being integrated is continuous and smooth over the interval.

In this specific task, we are using the same integral \( \int_{0}^{1} \sqrt{1-x^{2}} \, dx \) with \( n = 4 \). The interval is split into equal subintervals with a width of \( \Delta x = 0.25 \), just like in the Trapezoidal Rule.

Simpson’s Rule utilizes both the endpoints and the midpoints of the subintervals by weighting them differently. We evaluate the function \( f(x) = \sqrt{1-x^2} \) at the same points as in the Trapezoidal Rule: \( x_0 = 0 \), \( x_1 = 0.25 \), \( x_2 = 0.5 \), \( x_3 = 0.75 \), and \( x_4 = 1 \).

• Apply the Simpson's formula: \( S_n = \frac{\Delta x}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4)] \)

After computing this, the result is rounded to match the precision criteria given in the problem.

Definite Integral

A definite integral is a fundamental concept in calculus that represents the area under the curve of a function over a specified interval. Unlike an indefinite integral which has a constant of integration, a definite integral produces a numerical value.

The integral \( \int_{0}^{1} \sqrt{1-x^{2}} \, dx \) represents the total area under the function \( f(x) = \sqrt{1-x^2} \), from \( x = 0 \) to \( x = 1 \). This function describes a semicircle of radius 1 with its center at the origin, thus the integral gives us part of the area of a circle.

The exact calculation of such integrals can sometimes be complex, and so numerical methods like the Trapezoidal Rule and Simpson's Rule provide approximate values that are often sufficiently accurate for practical purposes, particularly with sufficient subdivisions \( n \). Accurate calculation of these definite integrals is crucial in various fields like physics and engineering.