44–49. Areas of regions Find the area of the following regions.
The region inside the cardioid r=1+cosθ and outside the cardioid r=1−cosθ

7–8. Parametric curves and tangent lines

a. Eliminate the parameter to obtain an equation in x and y.

x = 4sin 2t, y = 3cos 2t, for 0 ≤ t ≤ π; t = π/6

10–12. Parametric curves

a. Eliminate the parameter to obtain an equation in x and y.

x = ln t, y = 8ln t², for 1 ≤ t ≤ e²; (1, 16)

53–57. Conic sections

b. Use analytical methods to determine the location of the foci, vertices, and directrices.

x² - y²/2 = 1

Conic parameters: A hyperbola has eccentricity e = 2 and foci (0, ±2). Find the location of the vertices and directrices.

10–12. Parametric curves

a. Eliminate the parameter to obtain an equation in x and y.

x = t² + 4, y = -t, for -2 < t < 0; (5, 1)

53–57. Conic sections

b. Use analytical methods to determine the location of the foci, vertices, and directrices.

4x² + 8y² = 16