Arc Length Integral Formula, Distance, Problems and Solutions - Calculus

In calculus, arc length is the length of a curve between two points, calculated by integrating a formula that sums infinitesimal segments of the curve. For a smooth function \(y=f(x)\) over an interval \([a,b]\), the arc length \(L\) is found by integrating the square root of \((1+[f^{\prime }(x)]^{2})\) with respect to \(x\) from \(a\) to \(b\): \(L=\int _{a}^{b}\sqrt{1+[f^{\prime }(x)]^{2}}\,dx\).

💡Deriving the Arc Length Formula
The formula is derived by approximating the curve with a series of tiny line segments and finding the limit as the number of segments approaches infinity.
• Divide the curve: into \(n\) small segments.
• Approximate the length: of each segment with the hypotenuse of a right triangle, where the legs are the change in \(x\) (\(\Delta x\)) and the change in \(y\) (\(\Delta y\)).
• Use the Pythagorean theorem: The length of each segment is \(\sqrt{(\Delta x)^{2}+(\Delta y)^{2}}\).
• Rewrite: this in terms of the derivative \(\frac{dy}{dx}\):
\(\sqrt{(\Delta x)^{2}(1+(\frac{\Delta y}{\Delta x})^{2})}=\Delta x\sqrt{1+(\frac{dy}{dx})^{2}}\).
• Sum the lengths: of all these segments to approximate the total arc length:
\(\sum \Delta x\sqrt{1+(\frac{dy}{dx})^{2}}\).
• Take the limit: as \(\Delta x\) approaches 0, transforming the sum into a definite integral:
\(\int _{a}^{b}\sqrt{1+(\frac{dy}{dx})^{2}}\,dx\).

💡Formulas for Different Cases
• For \(y=f(x)\): If the curve is defined as a function of \(x\), the arc length \(L\) from \(x=a\) to \(x=b\) is:
\(L=\int _{a}^{b}\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}\,dx\quad \text{or}\quad L=\int _{a}^{b}\sqrt{1+[f^{\prime }(x)]^{2}}\,dx\).
For \(x=g(y)\): If the curve is defined as a function of \(y\), the arc length \(L\) from \(y=c\) to \(y=d\) is:
\(L=\int _{c}^{d}\sqrt{1+\left(\frac{dx}{dy}\right)^{2}}\,dy\quad \text{or}\quad L=\int _{c}^{d}\sqrt{1+[g^{\prime }(y)]^{2}}\,dy\).
• For Parametric Curves: For a curve defined by parametric equations \(x=x(t)\) and \(y=y(t)\), the arc length \(L\) from \(t=a\) to \(t=b\) is:
\(L=\int _{a}^{b}\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}}\,dt\).

💡Worksheets are provided in PDF format to further improve your understanding:
• Questions Worksheet: https://drive.google.com/file/d/1GQtYzPXeZpvnm7EdpOAPfEyr-VSltAse/view?usp=drive_link
• Answers: https://drive.google.com/file/d/1urkbDfPaDPsymE8T58TZZJHgeUw-P-qx/view?usp=drive_link

💡Chapters:
00:00 Arc length integral
02:04 Worked example

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