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


x = 2t sin t - t² cos t, y = 2t cos t + t² sin t; 0 ≤ t ≤ π

73–76. Tangent lines Find an equation of the line tangent to the curve at the point corresponding to the given value of t.

x=t ²−1, y=t ³ +t; t=2

How does the eccentricity determine the type of conic section?

84. Arc length for polar curves: Prove that the length of the curve r = f(θ) for α ≤ θ ≤ β is

L = ∫(α to β) √(f(θ)² + f'(θ)²) dθ.

45–60. Areas of regions Find the area of the following regions.

The region inside the rose r = 4 sin 2θ and inside the circle r = 2

15–22. Sets in polar coordinates Sketch the following sets of points.

1 < r < 2 and π/6 ≤ θ ≤ π/3

31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.

x=tan t, y=sec ² t−1