Problem 18 - Calculus Single Variable

Question

Calculate the length of the given parametric curve. $$ x=5 t-7 \quad y=12 t \quad 2 \leq t \leq 7 $$

Step-by-Step Solution

Answer

The length of the curve is 65 units.

Step 1: Understand the Parametric Equations

The curve is defined by the parametric equations \(x = 5t - 7\) and \(y = 12t\), where the parameter \(t\) ranges from 2 to 7. This defines a segment of the curve for \(t\) between these values.

Step 2: Derive the Differentials

Differentiate each parametric equation with respect to \(t\). For the first equation, \(\frac{dx}{dt} = 5\). For the second equation, \(\frac{dy}{dt} = 12\). These derivatives will help us determine the infinitesimal arc length \(ds\).

Step 3: Calculate the Differential Arc Length

The differential arc length \(ds\) is given by \[d s = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \]Substitute the derivatives, \[d s = \sqrt{5^2 + 12^2} \, dt = \sqrt{25 + 144} \, dt = 13 \, dt\]

Step 4: Integrate to Find Total Arc Length

To find the total length of the curve, integrate \(ds\) from \(t = 2\) to \(t = 7\):\[\text{Length} = \int_{2}^{7} 13 \, dt\]. This evaluates to \[13[t]\Big|_2^7 = 13(7) - 13(2) = 65\].

Step 5: Conclude the Result

The total length of the curve from \(t = 2\) to \(t = 7\) is 65 units. This is the final result of the integration.

Key Concepts

Parametric Equations

Parametric equations are a way of defining a curve in terms of an independent parameter, often represented as \( t \). In this context, each equation describes one coordinate of the curve. For example, an equation like \( x = 5t - 7 \) tells us how the \( x \)-coordinate changes with \( t \), while \( y = 12t \) describes the corresponding \( y \)-coordinates. This method is particularly useful when dealing with more complex shapes and movements that cannot be easily captured by a simple \( y = f(x) \) form.
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With parametric equations, the curve is traced out as \( t \) varies over a specified range. Here, \( t \) varies from 2 to 7, limiting the curve's exploration to these values. It is crucial to examine the range of \( t \) since it defines the segment of the curve we're interested in analyzing.

Differentiation

Differentiation is crucial when working with parametric equations because it allows us to calculate how each parameter changes. The process involves taking the derivative of each parametric equation with respect to \( t \).
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For our equations, differentiating \( x = 5t - 7 \) with respect to \( t \) gives \( \frac{dx}{dt} = 5 \), while differentiating \( y = 12t \) yields \( \frac{dy}{dt} = 12 \). These derivatives help us understand the rate of change of our variables. Knowing these rates is essential for calculating the arc length and other geometric properties of the curve.
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Through differentiation, we acquire insights into how quickly the curve "moves" horizontally and vertically as \( t \) changes. This knowledge sets the stage for the next step in determining the actual length.

Integral Calculus

Integral calculus is the branch of mathematics involved in calculating the areas under curves, volumes, and in this case, arc lengths. Once we have established the rates of change from differentiation, we use integration to accumulate these changes over the range of \( t \).
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First, we find the differential arc length \( ds \) using the formula:

• \( ds = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \)

Substituting in our derivatives, we get:
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\( ds = \sqrt{5^2 + 12^2} \, dt = \sqrt{25 + 144} \, dt = 13 \, dt \).
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This formula represents an infinitesimal segment of the curve length. Integral calculus now plays its role by summing these infinitesimal pieces to find the total length.

Curve Length Calculation

Calculating the length of a curve defined by parametric equations involves integrating the differential element \( ds \) over the specified interval of \( t \).
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For our parametric curve, the length is computed by evaluating the integral:

• \[ \text{Length} = \int_{2}^{7} 13 \, dt \]

This integral sums up all the small arc lengths from \( t = 2 \) to \( t = 7 \). Evaluating this provides:

• \[ 13[t]\Big|_2^7 = 13(7) - 13(2) = 65 \]

This result means the total length of the curve segment over this interval is 65 units.
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The curve length calculation involves both differentiation and integration and illustrates how these mathematical concepts bridge to measure something continuous like a curve. This approach is fundamental in various fields, from physics for understanding motion paths, to engineering for designing curves.