Sliding Sine Window

Relevant wiki: Fundamental Theorem of Calculus

Letting $x(t)$ be the position of the lower-left corner of the square at time $t$ we have that $x(t) = t$ . Since the square and the blue region both have a "ceiling" of $1$ , the area $A$ inside both the blue region and the square is just the area inside the blue region from $t$ to $t + 1$ , i.e.,

$A = \displaystyle\int_{t}^{t + 1} \sin(\theta) d\theta$ for $t \in [0, \pi - 1]$ .

We are looking for $\dfrac{dA}{dt}$ when $t = 1$ , and by the Fundamental Theorem of Calculus we have that

$\dfrac{dA}{dt} = \sin(t + 1) - \sin(t)$ , which at $t = 1$ is $\sin(2) - \sin(1) = \boxed{0.0678}$ to 3 significant figures.

A wave entering a

If one were to take a more physical approach without calculus, imagine the square to be a pipe (with its axis parallel to the x-axis and a rectangular cross section) The sine function graph is a water wave entering it. The rate at which the blue region (water) inside the pipe changes is $\frac{WaterFlowingIn - WaterFlowingOut}{time} = \frac{\sin(2) dx - \sin(1) dx}{dt} = [\sin 2 - \sin 1] \times \frac{dx}{dt} = [sin 2 - sin 1] \times 1= 0.0678$