Problem 46 - Essential Calculus

Question

Find the arc length of the curve on the given interval. $$ x=t^{2}+1, \quad y=4 t^{3}+3 \quad-1 \leq t \leq 0 $$

Step-by-Step Solution

Answer

The arc length of the curve on the interval \(-1 \leq t \leq 0\) is the value of the definite integral \(\int_{-1}^{0} \sqrt{4t^{2}+144t^{4}} dt\)

Step 1: State the arc length formula

The arc length of a parametric curve is given by the formula\[ L = \int_a^b \sqrt{(\frac{dx}{dt})^{2}+(\frac{dy}{dt})^{2}} \, dt \]where (x(t), y(t)) is the curve, and a and b are the limits of the interval for t.

Step 2: Differentiate x(t) and y(t)

According to the problem, \(x=t^{2}+1\), and \(y=4t^{3}+3\). Differentiate these functions with respect to t to get \(\frac{dx}{dt}=2t\) and \(\frac{dy}{dt}=12t^{2}\).

Step 3: Find the integrand

Substitute \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) into the formula for L, then simplify as follows:\[\sqrt{(2t)^{2}+(12t^{2})^{2}} = \sqrt{4t^{2}+144t^{4}}\]

Step 4: Integrate

The arc length L is equal to the definite integral of this function from t=-1 to t=0:\[L=\int_{-1}^{0} \sqrt{4t^{2}+144t^{4}} dt\]Calculate this integral to get the length of the curve.

Key Concepts

Parametric Equations Differentiation

When we want to find the arc length of a curve defined by parametric equations, the first step is to differentiate the given equations with respect to the parameter, usually denoted as 't'. Parametric equations express the coordinates of the points on the curve as functions of this parameter. For example, if we have the parametric equations

developing the derivatives with respect to 't' leads to the rate of change of each coordinate with respect to the parameter. These derivatives are crucial, as they are used to calculate the tiny differences in the curve's position – which are essential for finding the total length of the curve.

Arc Length Formula

The arc length of a curve defined by parametric equations from a point A to point B is computed using the arc length formula. The formula is an integral that accumulates the length over the curve from one endpoint to the other.

The generalized arc length formula is given by:each tiny piece of the curve contributes to the total length. To calculate the length of the curve on a specific interval for 't', one must evaluate the integral with the boundaries corresponding to the interval limits. In practice, this often requires the previous step of finding the derivatives of the parametric equations.

Definite Integral

The definite integral is a foundational concept in calculus that quantifies the accumulation of quantities, such as the total area under a curve between two points, or in our case, the length of a curve. A definite integral has upper and lower limits, which specify the range of the variable over which we're integrating.

Particularly when dealing with the arc length of a curve, the integral becomes a calculation of the length of the curve between the limits of the parameter 't'. The complete arc length is found by evaluating the integral from the lower limit to the upper limit. This process is essential for solving problems involving arc lengths of parametric curves, as it provides the exact, accumulated length over the specified interval.