Theory and Examples


Volume Find the volume of the solid generated by revolving the region enclosed by the ellipse 9x² + 4y² = 36 about the y−axis.

Cartesian to Polar Coordinates

Find the polar coordinates, 0 ≤ θ ≤ 2π and r ≤ 0, of the following points given in Cartesian coordinates.

c. (−1, √3)

Polar Coordinates

Plot the following points, given in polar coordinates. Then find all the polar coordinates of each point.

a. (2, π/2)

Polar to Cartesian Coordinates

Find the Cartesian coordinates of the following points, given in polar coordinates.

c. (0, π/2)

Cartesian to Polar Coordinates

Find the polar coordinates, 0 ≤ θ < 2π and r ≥ 0, of the following points given in Cartesian coordinates.

b. (-3,0)

Shifting Conic Sections

You may wish to review Section 1.2 before solving Exercises 39-56.

The hyperbola (y²/4) − (x²/5) = 1 is shifted 2 units down to generate the hyperbola (y + 2)²/4 − x²/5 = 1.

a. Find the center, foci, vertices, and asymptotes of the new hyperbola.