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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.

Level 3

         

Lise jogs along the curve defined by \(f(x) =\dfrac{2(x - 1)^{\frac{3}{2}}}{3}\) from \((1, f(1))\) to \((4, f(4)).\) Steve jogs along the straight line connecting those two points. Steve and Lise both start from \(x = 1\) at the same time and Lise jogs at a speed of \(\frac{7}{\sqrt{3}} \mbox{ units}\mbox{/s}\). What is the speed at which Steve must run (in \(\mbox{ units}\mbox{/s}\)) so that he arrives at \((4, f(4))\) at the same time as Lise?

The coordinates of a dot \(P\) moving in the \(xy\) plane at time \(t\) are given by: \(x = 5e^{-t}\cos t\) and \(y = 5e^{-t} \sin t\). Let \(L_a\) be the distance traveled by \(P\) in the interval \(0 \leq t \leq a\). If \(\displaystyle \lim_{a \to \infty} L_a = z\), what is the value of \(z^2\)?

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

There exists a unique, positive-valued, non-constant, continuous and differentiable function \(y = f(x)\) such that

(i) over any specified interval, the area between \(f(x)\) and the \(x\)-axis is equal to the arclength of the curve, and

(ii) \(f(0) = 1\).

If \(S = \displaystyle\int_{-1}^{2} f(x) dx\), then find \(\lfloor 1000S \rfloor\).

Consider the curve \(y={ e }^{ -x }\) in the first quadrant. Now it's rotated about the \(x\text{-axis}\) to obtain a solid of revolution. What is its surface area to 4 decimal places?

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