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Arc Length and Surface Area

Finding the perimeter of arbitrary curves and the area of 3D shapes foils the traditional tools of Geometry, and calls for the help of integrals and derivatives to make these calculations.

Problem Solving

         

What is the area of the surface obtained by rotating the circle \(x^2+y^2=15 ^2\) about the line \(y=15 ?\)

Hint: You can use \[\int \frac{du}{\sqrt{a^2-u^2}}=\sin^{-1} \frac{u}{a}+C,\ a > 0.\]

What is the length of the curve represented by the equation \(\displaystyle x^{2/3}+y^{2/3}=1?\)

What is the length of the curve \(\displaystyle y=\int_{3}^{x} \sqrt{t^3-1} \,dt\) in the domain \(1 \leq x \leq 4 ?\)

What is the area of the surface obtained by rotating the loop of the curve \(48y^2=x(16-x)^2\) about the \(x\)-axis?

A delivery drone flying at constant speed \(15 \text{ m/s}\) and constant height \(2700 \text{ m}\) toward a destination drops its goods. If the trajectory of the falling goods until it hits the ground can be described by the equation \[y=2700-\frac{x^2}{75},\] where \(x\) is the horizontal distance it travels and \(y\) is its height above the ground, what is the distance (not horizontal displacement) traveled by the goods until it hits the ground?

Note: You can use \(\displaystyle \int \sqrt{a^2+u^2}\,du=\frac{u}{2}\sqrt{a^2+u^2}+\frac{a^2}{2}\ln(u+\sqrt{a^2+u^2})+C.\)

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