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

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

Second derivative Assume a curve is given by the parametric equations x=f(t) and y=g(t), where f and g are twice differentiable. Use the Chain Rule to show that y″x=(fʹ(t)g″(t)−gʹ(t)f″(t))/(fʹ(t))³.  

Length in Polar Coordinates

Find the lengths of the curves given by the polar coordinate equations in Exercises 51–54.

r = √(1 + cos 2θ), −π/2 ≤ θ ≤ π/2

Finding Parametric Equations and Tangent Lines

Find parametric equations for the given curve.

Line through (1,-2) with slope 3

Tangent Lines to Parametrized Curves

In Exercises 1−14, find an equation for the line tangent to the curve at the point defined by the given value of t. Also, find the value of d²y/dx² at this point.

x = sec² t − 1, y = tan t, t = −π/4

Tangent Lines to Parametrized Curves

In Exercises 1−14, find an equation for the line tangent to the curve at the point defined by the given value of t. Also, find the value of d²y/dx² at this point.

x = t + eᵗ, y = 1 − eᵗ, t = 0

Cycloid

a. Find the length of one arch of the cycloid x = a(t − sin t), y = a(1 − cos t).

Centroids

Find the coordinates of the centroid of the curve x = cos t, y = t + sin t, 0 ≤ t ≤ π.

Lengths of Curves

Find the lengths of the curves in Exercises 25–30.

x = cos t, y = t + sin t, 0 ≤ t ≤ π