Problem 14 - Thomas Calculus

Question

In Exercises \(11-14,\) find the arc length parameter along the curve from the point where \(t=0\) by evaluating the integral $$ s=\int_{0}^{t}|\mathbf{v}(\tau)| d \tau $$ from Equation \((3) .\) Then find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=(1+2 t) \mathbf{i}+(1+3 t) \mathbf{j}+(6-6 t) \mathbf{k}, \quad-1 \leq t \leq 0 $$

Step-by-Step Solution

Answer

The length of the curve is 7 units.

Step 1: Calculate the Velocity Vector

The velocity vector \( \mathbf{v}(t) \) is the derivative of the position vector \( \mathbf{r}(t) \). Differentiate each component of \( \mathbf{r}(t) \) with respect to \( t \). This gives: \[ \mathbf{v}(t) = \frac{d}{dt}(1+2t)\mathbf{i} + \frac{d}{dt}(1+3t)\mathbf{j} + \frac{d}{dt}(6-6t)\mathbf{k} = 2\mathbf{i} + 3\mathbf{j} - 6\mathbf{k} \]

Step 2: Calculate the Magnitude of the Velocity Vector

The magnitude of the velocity vector \( |\mathbf{v}(t)| \) is calculated as follows: \[ |\mathbf{v}(t)| = \sqrt{(2)^2 + (3)^2 + (-6)^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \]

Step 3: Set Up the Arc Length Integral

The arc length \( s \) from \( t=0 \) is given by the integral of the magnitude of the velocity vector: \[ s = \int_{0}^{t} |\mathbf{v}(\tau)| d\tau = \int_{0}^{t} 7 \, d\tau \]

Step 4: Evaluate the Arc Length Integral

Evaluate the integral to find the expression for arc length \( s \): \[ s = \int_{0}^{t} 7 \, d\tau = 7[\tau]_{0}^{t} = 7(t - 0) = 7t \]

Step 5: Find the Length of the Curve Portion

Substitute the values of \( t = -1 \) to \( t = 0 \) to find the total length of the indicated portion of the curve. Calculate \( s \) when \( t = 0 \) and \( t = -1 \): \[ s(t = 0) = 7 \times 0 = 0 \] \[ s(t = -1) = 7 \times (-1) = -7 \] The arc length or the length of the curve from \( t = -1 \) to \( t = 0 \) is \( |-7| = 7 \).

Key Concepts

Velocity Vector

When dealing with motion along a curve, understanding the velocity vector is crucial. The velocity vector is essentially the derivative of the position vector. It tells us how fast and in what direction an object is moving at any given moment. For our problem, the position vector is given as \( \mathbf{r}(t) = (1+2t)\mathbf{i} + (1+3t)\mathbf{j} + (6-6t)\mathbf{k} \). By differentiating each component with respect to \( t \), we obtain the velocity vector \( \mathbf{v}(t) = 2\mathbf{i} + 3\mathbf{j} - 6\mathbf{k} \). This new vector indicates that the object moves 2 units in the x-direction, 3 units in the y-direction, and -6 units in the z-direction per unit time. Grasping the dynamics embedded in the velocity vector provides insights into the object's movement along its path.

Magnitude of Velocity

The magnitude of the velocity vector gives us the speed of the moving object. It's like asking, "How fast is the object moving regardless of direction?" In mathematical terms, the magnitude of a vector \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} \) is found using the formula \( |\mathbf{v}| = \sqrt{a^2 + b^2 + c^2} \).For our velocity vector \( \mathbf{v}(t) = 2\mathbf{i} + 3\mathbf{j} - 6\mathbf{k} \), the magnitude is calculated as follows:

• Square each component: \( 2^2 = 4 \), \( 3^2 = 9 \), \((-6)^2 = 36 \)
• Add these squares together: \( 4 + 9 + 36 = 49 \)
• Take the square root: \( \sqrt{49} = 7 \)

The magnitude tells us that the object is moving at a constant speed of 7 units per unit time along the curve.

Arc Length Integral

The arc length integral helps us determine the total length traveled by the object along the curve. It's an integral of the magnitude of the velocity vector over a specific time interval. Therefore, it essentially adds up how far the object travels at each moment in time. Given the equation \( s = \int_{0}^{t} |\mathbf{v}(\tau)| d\tau \), we substitute the constant magnitude of the velocity, 7, to set up the integral:\[ s = \int_{0}^{t} 7 \, d\tau \]This integral simplifies greatly because the integrand is constant, leading to:\[ s = 7\tau \bigg|_{0}^{t} = 7t \]This result gives us a straightforward expression for the arc length, \( s = 7t \), highlighting the linear relationship between the time \( t \) and the arc length traveled.

Curve Length Calculation

Calculating the length of a specific portion of the curve involves evaluating the arc length expression over the indicated interval. In our problem, we seek to find the curve length from \( t = -1 \) to \( t = 0 \).Using the expression \( s = 7t \), we calculate:

• At \( t = 0 \), \( s = 7 \times 0 = 0 \)
• At \( t = -1 \), \( s = 7 \times (-1) = -7 \)

To find the actual length of the curve, we take the absolute difference between these two values, resulting in:\[ |s(t=0) - s(t=-1)| = |0 - (-7)| = 7 \]This calculation shows that the object travels a total length of 7 units along the curve between these two points in time.