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

Answer:

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) $s \approx 9.715$

Step-by-step 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:

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

$s \approx 9.715$