Centroids


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

Finding Cartesian from Parametric Equations

In Exercises 19–24, match the parametric equations with the parametric curves labeled A through F.

x = cos t, y = sin 3t

Finding Lengths of Polar Curves

Find the lengths of the curves in Exercises 21–28.

The curve r = cos³(θ/3), 0 ≤ θ ≤ π/4

Identifying Graphs

Match the parabolas in Exercises 1−4 with the following equations: x² = 2y, x² = −6y, y² = 8x, y² = −4x

Then find each parabola's focus and directrix.

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

Hyperbolas

Exercises 27-34 give equations for hyperbolas. Put each equation in standard form and find the hyperbola's asymptotes. Then sketch the hyperbola. Include the asymptotes and foci in your sketch.

8x² − 2y² = 16

Polar to Cartesian Equations

Replace the polar equations in Exercises 27–52 with equivalent Cartesian equations. Then describe or identify the graph.

r = 3 cos θ