Activity 6.1.4.
Each of the following questions involves the arc length along a curve.
(a)
Use the definition and appropriate computational technology to determine the arc length along \(y = x^2\) from \(x = -1\) to \(x = 1\text{.}\)
(b)
Find the arc length of \(y = \sqrt{4-x^2}\) on the interval \(-2 \le x \le 2\text{.}\) Find this value in two different ways: (a) by using a definite integral, and (b) by using a familiar property of the curve.
(c)
(d)
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?
(e)
A moving particle is traveling along the curve given by \(y = f(x) = 0.1x^2 + 1\text{,}\) 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 \(t = 0\text{,}\) the particle’s location is \((0,f(0))\text{.}\)
