Question

g For the curve parameterized by x(t) = 3 sin t, y(t) = 5 cost, for −π/4 ≤ t ≤ π/2: (a) Sketch the curve and the direction traced out as t increases. (b) Set up, but do not evaluate, the arc length of the curve. (c) Geometrically, and without solving an integral, estimate the length of the curve.

Answer 1

a) See the file below, b)
`s = \int\limits^(0.5\pi)_(-0.25\pi) {[\left( 3\cdot \cos t\right)^(2)+\left(-5\cdot \sin t \right)^(2)]} \, dx`, c)
`s \approx 9.715`

Explanation:

a) Points moves clockwise as t increases. See the curve in the file attached below. The parametric equations describe an ellipse.

b) The arc length formula is:

`s = \int\limits^(0.5\pi)_(-0.25\pi) {[\left( 3\cdot \cos t\right)^(2)+\left(-5\cdot \sin t \right)^(2)]} \, dx`

c) The perimeter of that arc is approximately:

`s \approx ((1)/(4) + (1)/(8))\cdot 2 \pi\cdot \sqrt{(3^(2)+5^(2))/(2) }`

`s \approx 9.715`