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

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

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

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

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

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

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

Find $\frac{d^{3}y}{d{x}^3}$ for the parametric curve at the given point.

$x=2t^3$, $y=t^4$, $t=1$