16. Parametric Equations & Polar Coordinates

81–88. Arc length Find the arc length of the following curves on the given interval.

x = eᵗ sin t, y = eᵗ cos t; 0 ≤ t ≤ 2π

81–88. Arc length Find the arc length of the following curves on the given interval.

x = sin t, y = t - cos t; 0 ≤ t ≤ π/2

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

e. There are two points on the curve x=−4 cos t, y=sin t, for 0≤t≤2π, at which there is a vertical tangent line.

22–23. Arc length Find the length of the following curves.

x = cos 2t, y = 2t - sin 2t; 0 ≤ t ≤ π/4

Find the area of the region bounded by the astroid x = cos³ t, y = sin³ t, for 0 ≤ t ≤ 2π

19–20. Area bounded by parametric curves Find the area of the following regions. (Hint: See Exercises 103–105 in Section 12.1.) The region bounded by the y-axis and the parametric curve

The region bounded by the x-axis and the parametric curve x=cost, y=sin2t, for 0≤t≤π/2