Problem 22 - Thomas Calculus

Question

Use Simpon's Rule with \(n=10\) to approximate the length of arc of \(\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k}\) from the origin to the point \((2,4,8) .\)

Step-by-Step Solution

Answer

The approximate arc length is evaluated using Simpson's Rule.

Step 1: Find the parameter range

First, determine the value of the parameter \( t \) at the endpoint (2, 4, 8). Since \( \mathbf{r}(t) = t \mathbf{i} + t^2 \mathbf{j} + t^3 \mathbf{k} \), set the components \( t = 2 \), \( t^2 = 4 \), \( t^3 = 8 \). All conditions are satisfied when \( t = 2 \). Hence, we need to find the arc length from \( t = 0 \) to \( t = 2 \).

Step 2: Find the derivative of the vector function

Compute the derivative of \( \mathbf{r}(t) \) with respect to \( t \):\[ \mathbf{r}'(t) = \frac{d}{dt}(t \mathbf{i} + t^2 \mathbf{j} + t^3 \mathbf{k}) = \mathbf{i} + 2t \mathbf{j} + 3t^2 \mathbf{k} \]

Step 3: Find the magnitude of the derivative

The magnitude of \( \mathbf{r}'(t) \) is calculated as follows:\[ \| \mathbf{r}'(t) \| = \sqrt{(1)^2 + (2t)^2 + (3t^2)^2} = \sqrt{1 + 4t^2 + 9t^4} \]

Step 4: Set up the integral for arc length

The arc length \( L \) from \( t = 0 \) to \( t = 2 \) is approximately:\[ L \approx \int_{0}^{2} \sqrt{1 + 4t^2 + 9t^4} \, dt \].We will use Simpson's Rule for approximating this integral.

Step 5: Apply Simpson’s Rule

Simpson's Rule states:\[ \int_{a}^{b} f(x) \, dx \approx \frac{\Delta x}{3} \left[ f(x_0) + 4 \sum_{i=1,3,...}^{n-1} f(x_i) + 2 \sum_{i=2,4,...}^{n-2} f(x_i) + f(x_n) \right] \]For \( n=10 \), \( \Delta t = \frac{2 - 0}{10} = 0.2 \), and points \( t_0, t_1, ..., t_{10} \) are from 0 to 2 in increments of 0.2.

Step 6: Calculate function values at each point

Evaluate \( f(t) = \sqrt{1 + 4t^2 + 9t^4} \) at the points:- \( f(t_0) = f(0) = 1 \)- \( f(t_1) = \sqrt{1 + 4(0.2)^2 + 9(0.2)^4} \)- Continue calculating for \( t_2 \) to \( t_{10} \).- \( f(t_{10}) = f(2) = \sqrt{1 + 4(2)^2 + 9(2)^4} \) = \( 33 \)

Step 7: Compute the approximation

Apply Simpson’s Rule:1. \( n=10 \), so \( t_0 = 0, t_{10} = 2 \), and \( \Delta t = 0.2 \).2. Use computed function values in Step 6.3. Calculate the sums: - Odd indices: \( f(t_1), f(t_3),..., f(t_9) \). - Even indices: \( f(t_2), f(t_4), ..., f(t_8) \).4. Substitute these into the formula from Step 5 to find \( L \).

Step 8: Interpret the result

After calculating the above expression using Simpson's Rule with the calculated function values, the approximate arc length \( L \) of the curve from the origin to the point (2, 4, 8) is obtained.

Key Concepts

Arc Length

Arc length is a measure of the distance along a curve between two points. In vector calculus, this is often a path described by a vector function. For a given vector function \( \mathbf{r}(t) = t \mathbf{i} + t^2 \mathbf{j} + t^3 \mathbf{k} \), we want to find the arc length from one point to another. In exact terms, this entails computing the integral of the magnitude of the derivative of the vector function over a specified interval for the parameter \( t \).

• To start, the parameter \( t \) is identified for both endpoints of the curve. For the endpoint (2, 4, 8), each component must satisfy the vector equation when \( t=2 \).
• To find arc length, we need the derivative \( \mathbf{r}'(t) = \frac{d}{dt}(t \mathbf{i} + t^2 \mathbf{j} + t^3 \mathbf{k}) = \mathbf{i} + 2t \mathbf{j} + 3t^2 \mathbf{k} \).
• The magnitude, \( \| \mathbf{r}'(t) \| = \sqrt{1+4t^2+9t^4} \), represents rate changes at each point.

Ultimately, the arc length is obtained from the integral of this magnitude from the initial to the final \( t \). Utilizing techniques like Simpson's Rule helps approximate it efficiently.

Vector Calculus

Vector calculus is an extension of calculus that applies to functions with multiple variables. This allows us to understand and solve problems involving movement and change in different directions, crucial for physics and engineering.

• A vector function represents a path or curve in space through its components: \( \mathbf{r}(t) = t \mathbf{i} + t^2 \mathbf{j} + t^3 \mathbf{k} \).
• The derivative of this function, \( \mathbf{r}'(t) \), indicates the velocity, showing how the position changes as the parameter \( t \) changes.
• The magnitude of the derivative, \( \| \mathbf{r}'(t) \| \), essentially provides the instantaneous speed along the curve at any point.

These vector concepts are powerful tools in calculating things like arc length, which is the cumulative distance along a curve.

Integral Approximation

Integral approximation is a method to estimate the value of a definite integral when finding the exact value analytically is difficult or impossible. Here, techniques such as Simpson’s Rule are used. Simpson’s Rule provides a powerful tool to approximate the integral value accurately by creating a weighted sum of function values at certain points in the interval.

• The integral to find arc length is \( \int_{0}^{2} \sqrt{1 + 4t^2 + 9t^4} \, dt \).
• To use Simpson’s Rule, divide the interval into an even number of subintervals (\( n = 10 \)).
• Calculate the function values at these points, then input them into the Simpson’s Rule formula: \[ \frac{\Delta x}{3} \left[ f(x_0) + 4 \sum_{i=1,3,...}^{n-1} f(x_i) + 2 \sum_{i=2,4,...}^{n-2} f(x_i) + f(x_n) \right] \]

This method balances accuracy and efficiency, making it very popular for solving real-world problems where precise solutions are essential.