Problem 51 - Calculus Single Variable

Question

Let \(N\) be an even integer. From equation \((5.8 .9)\) we may deduce that one of the estimates \(\mathcal{T}_{N / 2}\) and \(\mathcal{M}_{N / 2}\) is greater than or equal to \(\mathcal{S}_{N},\) and the other is less than or equal to \(\mathcal{S}_{N}\). In each of Exercises \(51-54,\) calculate \(\mathcal{S}_{N}, \mathcal{M}_{N / 2},\) and \(\mathcal{T}_{N / 2}\) for the given even value of \(N\) and verify this deduction. $$ \int_{0}^{4} x^{4} d x \quad N=4 $$

Step-by-Step Solution

Answer

Simpson's Rule \( \mathcal{S}_4 = 204.8 \), Midpoint Rule \( \mathcal{M}_2 = 82 \), Trapezoidal Rule \( \mathcal{T}_2 = 136 \). Simpson's Rule consistently provides a more accurate approximation or greater value.

Step 1: Understanding the Problem

You are asked to find the integral \( \int_0^4 x^4 \, dx \) with \( N = 4 \) and verify the relation between \( \mathcal{S}_N, \mathcal{M}_{N/2}, \) and \( \mathcal{T}_{N/2} \) for this integral using the trapezoidal rule \( \mathcal{T} \), midpoint rule \( \mathcal{M} \), and Simpson's rule \( \mathcal{S} \).

Step 2: Calculate Exact Integral \( \mathcal{I} \)

Find the exact value of \( \int_0^4 x^4 \, dx \) by performing the following integration: \[\int_0^4 x^4 \, dx = \left[ \frac{x^5}{5} \right]_0^4 = \frac{4^5}{5} - \frac{0^5}{5} = \frac{1024}{5} = 204.8.\]

Step 3: Calculate \( \mathcal{S}_N \) Using Simpson's Rule

Simpson's Rule for \( N = 4 \) is given by: \(\mathcal{S}_N = \frac{h}{3} \left( f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4) \right),\) where \( h = 1 \). Calculate \( \mathcal{S}_N = \frac{1}{3}(f(0) + 4f(1) + 2f(2) + 4f(3) + f(4)) = \frac{1}{3}(0 + 4 + 32 + 324 + 1024) = \frac{1632}{3} = 204.8. \)

Step 4: Calculate \( \mathcal{M}_{N/2} \) Using Midpoint Rule

The midpoint rule for \( N/2 = 2 \) is: \( \mathcal{M}_{N/2} = f(1) + f(3) = 1^4 + 3^4 = 1 + 81 = 82. \)

Step 5: Calculate \( \mathcal{T}_{N/2} \) Using Trapezoidal Rule

The trapezoidal rule for \( N/2 = 2 \) is: \[\mathcal{T}_{N/2} = \frac{f(0) + f(2) + f(4) }{2} = \frac{0 + 16 + 256}{2} = \frac{272}{2} = 136.\]

Step 6: Verify the Deductions

Verify the deduction \( \mathcal{T}_{N/2} < \mathcal{S}_N \) and \( \mathcal{M}_{N/2} < \mathcal{S}_N \). Here, \( 82 < 204.8 \) and \( 136 < 204.8 \). The condition now becomes a straightforward verification since one is meant to be greater or less as stated from the equation \( (5.8 .9) \). Ensure this aligns through comparison.

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a simple yet efficient technique in numerical integration. It is used to estimate the integral of a function. Picture the curve of a function on a graph. The trapezoidal rule works by approximating this curve with a series of straight line segments between the endpoints of subintervals in the range you're interested in. These straight segments form trapezoids, hence the name.

To apply this rule, you need to break down the interval, in this case, from 0 to 4, into smaller sub-intervals. For this problem, we have chosen 2 as the number of subintervals (as already calculated in the step solution with \(N/2=2\)). The endpoints are then used alongside function evaluations to form the trapezoids:

• You calculate the area of each trapezoid and sum these areas to get an approximation of the overall integral.
• The trapezoidal rule approximates the given integral: \( \mathcal{T}_{N/2} = \frac{f(0) + f(2) + f(4)}{2} = 136 \).

The trapezoidal rule is simple but less accurate compared to rules using higher-order polynomials, like Simpson's Rule.

Midpoint Rule

The Midpoint Rule offers another method for estimating integrals using numerical integration. It differs from the trapezoidal rule by not using the endpoints directly but rather focusing on the midpoints of the subintervals.

Imagine dividing the range from 0 to 4 into equal segments. Instead of using the edges to form shapes, consider the middle point of each segment. By plugging in this midpoint into your function, you determine the height of a rectangle, whose width is the length of the segment:

• Summing up the areas of these rectangles gives you an approximation of the integral.
• For our example, the calculation is done over segments \([0,2]\) and \([2,4]\), with midpoints at \(1\) and \(3\): \( \mathcal{M}_{N/2} = f(1) + f(3) = 82 \).

This method often gives results that are quite close to the exact value between trapezoidal and Simpson's, striking a balance between simplicity and accuracy.

Numerical Integration

Numerical integration involves using algorithms to approximate the value of an integral. These methods are especially useful when the integral is difficult or impossible to solve analytically.

In numerical integration:

• We choose a method that fits the problem and desired accuracy.
• Commonly used techniques include the Trapezoidal Rule, Midpoint Rule, and Simpson's Rule.
• These methods break down the integral into simpler parts that are easily computed.

Each method has different approaches and levels of accuracy. Simpson's Rule, for instance, uses parabolic segments to approximate the curve. Each technique offers a distinct balance between complexity and precision. Deciding which method to use often depends on the function's behavior and how fine of an approximation is required. Always remember, the objective is to estimate the true integral's value as closely as possible with minimal computational effort.