16. Parametric Equations & Polar Coordinates

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

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

Centroids

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

77–80. Slopes of tangent lines Find all points at which the following curves have the given slope.

x = 4 cos t, y = 4 sin t; slope = 1/2

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

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=cos t+t sin t,y=sin t−t cos t; t=π/4

Write the equation of the tangent line in cartesian coordinates for the given parameter $t$.

$x=8\cos t$, $y=6\sin t$, $t=\frac{\pi }{4}$

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.  

Find $\frac{d^{2}y}{d{x}^2}$ for the parametric curve at the given point.

$x=8\cos t$, $y=6\sin t$, $t=\frac{\pi }{4}$

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

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

77–80. Slopes of tangent lines Find all points at which the following curves have the given slope.

x = 2 cos t, y = 8 sin t; slope = -1

Surface Area

Find the areas of the surfaces generated by revolving the curves in Exercises 31-34 about the indicated axes.

x = t + √2, y = (t²/2) + √2t, −√2 ≤ t ≤ √2; y−axis

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

Lengths of Curves

Find the lengths of the curves in Exercises 13–19.

x = 5 cos t − cos 5t, y = 5 sin t − sin 5t, 0 ≤ t ≤ π/2