8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled

4 min read•february 15, 2024

AP Calculus AB/BC ♾️

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The Arc Length of a Smooth, Planar Curve and Distance Traveled

In this section of the AP Calculus BC course, students learn how to find the length of a smooth, planar curve, also known as the arc length. This topic is also covered in the context of finding the distance traveled along a curve.

Here are the steps to find the arc length of a smooth, planar curve:

Parametrize the curve: The first step is to parametrize the curve in terms of a single variable t. This means writing the curve in terms of x(t) and y(t) where x and y are functions of t.
Find the arc length function: The next step is to find the arc length function, also known as the length element. This is done by taking the square root of the sum of the squares of the derivatives of x and y with respect to t. The arc length function is given by the formula: √(dx/dt)^2 + (dy/dt)^2.
Evaluate the definite integral: The final step is to evaluate the definite integral of the arc length function from the initial value of t to the final value of t. The definite integral gives us the length of the curve.

To find the distance traveled along a curve, students need to use the following steps:

Parametrize the curve: As with the arc length, the first step is to parametrize the curve in terms of a single variable t.
Find the velocity function: The next step is to find the velocity function. This is done by taking the derivative of the position function with respect to time.
Evaluate the definite integral: The final step is to evaluate the definite integral of the velocity function from the initial time to the final time. The definite integral gives us the distance traveled along the curve.

Here are some examples:

Example 1: Consider the curve defined by x(t) = t^3 and y(t) = t^2. To find the arc length of this curve, first parametrize the curve. Then, find the arc length function by taking the square root of the sum of the squares of the derivatives of x and y with respect to t. Finally, evaluate the definite integral of the arc length function from 0 to 1 to find the length of the curve.

Example 2: Consider a particle moving along the curve defined by x(t) = t^2 and y(t) = t^3. To find the distance traveled by the particle along the curve, first parametrize the curve. Then, find the velocity function by taking the derivative of the position function with respect to time. Finally, evaluate the definite integral of the velocity function from 0 to 1 to find the distance traveled by the particle along the curve.

Example 3: Consider a particle moving along the curve defined by x(t) = cos(t) and y(t) = sin(t). To find the distance traveled by the particle along the curve, first parametrize the curve. Then, find the velocity function by taking the derivative of the position function with respect to time. Finally, evaluate the definite integral of the velocity function from 0 to π/2 to find the distance traveled by the particle along the curve.

Example 4: Consider the curve represented by the equation y = x^3, where x is between 0 and 2. To find the arc length of this curve, we need to first find the derivative of the curve, which is y' = 3x^2. Next, we use the formula for arc length, L = ∫√(1 + (y')^2) dx from x = 0 to x = 2. After evaluating the integral, we find that the arc length of the curve is approximately 3.72 units.

Example 5: A particle moves along the curve represented by the equation y = x^2 + 1, where x is between 0 and 2. To find the distance traveled by the particle, we use the formula for arc length, L = ∫√(1 + (y')^2) dx from x = 0 to x = 2. After evaluating the integral, we find that the distance traveled by the particle is approximately 5.83 units.

The Arc Length of a Smooth, Planar Curve and Distance Traveled is a crucial topic in AP Calculus BC that involves finding the length of a curve or the distance traveled by a particle along a curve. To find the arc length of a curve, students need to find the derivative of the curve and use the formula for arc length, L = ∫√(1 + (y')^2) dx. To find the distance traveled by a particle, students use the same formula. Evaluating these integrals requires a good understanding of derivatives, integrals, and the formula for arc length. With proper practice and application, students can become proficient in this topic and excel in the AP Calculus BC exam.

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