Consider The Parametric Curve Given By The Equations Xt
Consider The Parametric Curve Given By The Equa Tionsxt
Consider the following assignment questions:
1. Consider the parametric curve given by the equations x(t) = t² - 10t + 31 and y(t) = t² - 10t + 21. How many units of distance are covered by the point P(t) = (x(t), y(t)) between t = 0 and t = 7?
2. Find the length of the parametrized curve given by x(t) = 0t³ - 12t² + 12t and y(t) = -4t³ + 6t² + 9t, for t between 0 and 1. (Hint: ds/dt is a quadratic polynomial with integer coefficients.)
3. Consider the parametric curve x(θ) = 2(cosθ + θsinθ) and y(θ) = 2(sinθ - θcosθ). What is the length of the curve between θ = 0 and θ = 18π?
4. The curve defined by x(θ) = 2cos³θ and y(θ) = 2sin³θ is called an astroid. Find the total arc length of this astroid.
5. Determine the area of the region enclosed by the astroid x(θ) = 7cos³θ and y(θ) = 7sin³θ.
6. The parametric curve x(t) = 9cos(t) and y(t) = 4sin(2t) intersects itself at a point P = (x₀, y₀) where x₀ = and y₀ =. The slopes of the 2 tangents at this point are and .
7. Find the area of the surface obtained by rotating the curve x = 4cos³θ, y = 4sin³θ, 0 ≤ θ ≤ π/2 about the y-axis. Surface area = .
8. Find the area of the surface obtained by rotating the curve x = 2t - (2/3)t³, y = 2t², 0 ≤ t ≤ 1 about the x-axis. Surface area = .
9. Decide if the points given in polar coordinates are the same. If so, enter T. If not, enter F. (9, π/3), (-9, -π), 5π/4; (0, 9π), (0, 7π/4); (1, 141π/4), (-1, π), 26π/3; (-9, -π), 4π; (-9, 4π).
10. For each set of Cartesian coordinates (x, y), match the equivalent set of Polar coordinates (r, θ).
11. Determine the area of the region enclosed by one loop of the three-leaved rose r = 3cos(3θ).
12. Determine the area of the region enclosed by the limacon r = 8 + 5sinθ.
13. Determine the length of the spiral r = 4θ for θ between 2 and 8.
14. Determine the total arc length of the cardioid r = 7 + 7sinθ.
Please use this information to prepare your comprehensive academic paper.