Activity 6.1.4

Each of the following questions involves the arc length along a curve.

1. Use the definition and appropriate computational technology to determine the arc length along $y = x^2$ from $x = -1$ to $x = 1$.

2. Find the arc length of $y = \sqrt{4-x^2}$ on the interval $-2 \le x \le 2$. Find this value in two different ways: (a) by using a definite integral, and (b) by using a familiar property of the curve.

3. Determine the arc length of $y = xe^{3x}$ on the interval $[0,1]$.

4. Will the integrals that arise calculating arc length typically be ones that we can evaluate exactly using the First FTC, or ones that we need to approximate? Why?

5. A moving particle is traveling along the curve given by $y = f(x) = 0.1x^2 + 1$, and does so at a constant rate of 7 cm/sec, where both $x$ and $y$ are measured in cm (that is, the curve $y = f(x)$ is the path along which the object actually travels; the curve is not a “position function”). Find the position of the particle when $t = 4$ sec, assuming that when $t = 0$, the particle's location is $(0,f(0))$.