Parametric Curve, Arc Length Formula, Integral, Visual Proof, Practice Problems - Calculus

The arc length (\(L\)) of a parametric curve given by \(x(t)\) and \(y(t)\) for \(t\) from \(a\) to \(b\) is found by integrating the square root of the sum of the squares of the derivatives of the x and y components with respect to \(t\), from \(a\) to \(b\), using the formula: \(L=\int _{a}^{b}\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}}\,dt\).

💡Here's how to calculate the arc length:
• Find the derivatives: of the parametric equations with respect to \(t\). Let \(x^{\prime }(t)=\frac{dx}{dt}\) and \(y^{\prime }(t)=\frac{dy}{dt}\).
• Square the derivatives: and add them together: \((x^{\prime }(t))^{2}+(y^{\prime }(t))^{2}\).
• Take the square root: of this sum: \(\sqrt{(x^{\prime }(t))^{2}+(y^{\prime }(t))^{2}}\).
• Integrate: the result with respect to \(t\) over the given interval, from \(a\) to \(b\).

💡Example:
Consider the parametric curve \(x(t)=t^{2}\) and \(y(t)=t^{3}\) for \(t\) from \(0\) to \(1\).
• Derivatives: \(\frac{dx}{dt}=2t\) and \(\frac{dy}{dt}=3t^{2}\).
• Square and add: \((2t)^{2}+(3t^{2})^{2}=4t^{2}+9t^{4}\).
• Integrate: \(L=\int _{0}^{1}\sqrt{4t^{2}+9t^{4}}\,dt=\int _{0}^{1}t\sqrt{4+9t^{2}}\,dt\).
Using a u-substitution (\(u=4+9t^{2}\)), the integral evaluates to \(\frac{1}{27}(13\sqrt{13}-8)\).

💡Why use this formula?
• It is derived from the Pythagorean theorem, where a small segment of the curve \(\Delta L\) is approximated by the hypotenuse of a small right triangle with sides \(\Delta x\) and \(\Delta y\). As \(\Delta t\rightarrow 0\), this becomes \(dL=\sqrt{(dx)^{2}+(dy)^{2}}\).
The arc length integral sums up these infinitesimal lengths to find the total length of the curve.
• It is especially useful for curves that cannot be easily expressed as a function of \(y\) in terms of \(x\), such as spirals.

💡Worksheets are provided in PDF format to further improve your understanding:
• Questions Worksheet: https://drive.google.com/file/d/1tolpJ9mIz-hrSc3BJyC0piUzPQ8cKD3b/view?usp=drive_link
• Answers: https://drive.google.com/file/d/1tiM6iM6LTV0bkAhLvC8ij3bjvbloV5Jj/view?usp=drive_link

💡Chapters:
00:00 Parametric arc length integral
01:56 Worked example

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