Parametric equations define $x$ and $y$ coordinates in terms of a third variable, often denoted as $t$. To find the arc length of a parametric curve, we use an integral that sums up infinitesimal lengths along the curve. The formula involves the derivatives of $x(t)$ and $y(t)$ with respect to $t$. Understanding this concept allows us to calculate the length of complex curves that are difficult to express in standard Cartesian form.
The arc length $L$ of a parametric curve defined by $x = f(t)$ and $y = g(t)$ from $t = a$ to $t = b$ is given by the integral: $L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$
Match each term with its definition:
| Term | Definition |
|---|---|
| 1. Parametric Curve | A. The rate of change of $y$ with respect to $t$ |
| 2. Arc Length | B. A curve defined by equations where $x$ and $y$ are functions of a third variable. |
| 3. $\frac{dx}{dt}$ | C. The rate of change of $x$ with respect to $t$ |
| 4. $\frac{dy}{dt}$ | D. A curve defined by equations where $x$ and $y$ are functions of $t$ |
| 5. Integral | E. The length along a curve. |
Correct Matches: 1-B, 2-E, 3-C, 4-A, 5-The process of measuring an area or volume.
The formula for arc length of a parametric curve involves finding the ________ of the derivatives of $x(t)$ and $y(t)$, squaring them, and then taking the ________ ________. We then ________ this expression with respect to ________ from $a$ to $b$. The result gives us the total ________ ________.
Answer: square, square root, integrate, $t$, arc length
Explain in your own words why it's necessary to use parametric equations to find the arc length of certain curves, and provide an example of a curve where parametric equations are particularly useful.
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