Calculus
# Arc Length and Surface Area

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

$y = f(x)$ such that

There exists a unique, positive-valued, non-constant, continuous and differentiable function(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 $\lfloor 1000S \rfloor$.

$y={ e }^{ -x }$ 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?

Consider the curve