Calculus

# Arc Length and Surface Area: Level 3 Challenges

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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