Find the length of the loop of the given curve. x = 12t − 4t3, y = 12t2

1. (1 pt) Consider the parametric curve given by the equa- tions

x(t) = t2−10t +31 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. (1 pt) Find the length of the parametrized curve given by

x(t) = 0t3−12t2 +12t, y(t) =−4t3 +6t2 +9t,

for t between 0 and 1: Hint: dsdt is a quadratic polynomial with integer coefficients.

3. (1 pt) Consider the parametric curve

x(θ) = 2(cosθ+θsinθ)

y(θ) = 2(sinθ−θcosθ) What is the length of the curve between θ = 0 and θ = 18 π?

4. (1 pt) The curve defined by:

x(θ) = 2cos3 θ

y(θ) = 2sin3 θ is called an astroid (sketch it to determine why!) Find the total arc length of this astroid:

5. (1 pt) Determine the area of the region enclosed by the astroid

x(θ) = 7cos3 θ y(θ) = 7sin3 θ

6. (1 pt) The parametric curve

x(t) = 9cos(t)

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. (1 pt) Find the area of the surface obtained by rotating the

curve of parametric equations

x = 4cos3 θ, y = 4sin3 θ, 0≤ θ≤ π/2

about the y axis. Surface area =

8. (1 pt) Find the area of the surface obtained by rotating the curve of parametric equations:

x = 2t− 2 3

t3, y = 2t2, 0≤ t ≤ 1

about the x - axis. Surface Area =

9. (1 pt) Decide if the points given in polar coordinates are the same. If so, enter T. If not, enter F. (9, π3 ),(−9,

−π 3 )

(2, 59π4 ),(2,− 59π

4 ) (0,9π),(0, 7π4 ) (1, 141π4 ),(−1,

π 4 )

(9, 26π3 ),(−9, −π 3 )

(9,4π),(−9,4π)

10. (1 pt) For each set of Cartesian coordinates (x,y), match the equivalent set of Polar coordinates (r,θ).

1. ( 8.5 , 3.7 ) 2. ( -0.199999999999999 , 10 ) 3. ( -0.399999999999999 , 9 ) 4. ( 0.100000000000001 , 6.3 )

A. (9.0089,1.6152) B. (9.2704,0.4106) C. (10.0020,1.5908) D. (6.3008,1.5549)

11. (1 pt) For each set of Polar coordinates in 1. to 6., match the approximately equivalent set of Cartesian coordinates in A. to F.

1. (10.1119,1.4219) 2. (8.7321,1.3397) 3. (9.4493,0.8828) 4. (2.4166,1.1442) 5. (11.8004,0.6350) 6. (4.7424,2.4009)

A. (−3.5,3.2) B. (6,7.3) C. (1.5,10) D. (9.5,7) E. (1,2.2) F. (2,8.5)

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12. (1 pt) Determine the area of the region enclosed by one loop of the three-leaved rose

r = 3cos(3θ)

13. (1 pt) Determine the area of the region enclosed by the limaçon

r = 8+5sinθ

14. (1 pt) Determine the length of the spiral

r = 4θ for θ between 2 and 8.

15. (1 pt) Determine the total arc length of the cardioid

r = 7+7sinθ

Generated by the WeBWorK system c©WeBWorK Team, Department of Mathematics, University of Rochester

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