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AP Calculus Chapter 10 Test Bank Mr. Surowski
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AP Calculus Testbank(Chapter 10)
(Mr. Surowski)
Part I. Multiple-Choice Questions
1. The graph in the xy-plane represented by x = 3 sin t and y =
2 cos t is
(A) a circle (B) an ellipse (C) a hyperbola (D) a parabola (E) a line
2. If a particle moves in the xy-plane so that at time t > 0 its po-
2 3
sition vector is (et , e−t ), then its velocity vector at time t = 3
is
(A) (ln 6, ln(−27))
(B) (ln 9, ln(−27))
(C) (e9 , e−27 )
(D) (6e9 , −27e−27 )
(E) (9e9 , −27e−27 )
3. A particle moves along a path described by x = cos3 t and y =
sin3 t. The distance that the particle travels along the path from
π
t = 0 to t = is
2
(A) 0.75 (B) 1.50 (C) 0 (D) −3.50 (E) −0.75
4. The area enclosed by the polar equation r = 4 + cos θ, for 0 ≤
θ ≤ 2π, is
9π 33π 33π
(A) 0 (B) (C) 18π (D) (E)
2 2 4
dy
5. If, for t > 0, x = t2 and y = cos t2 , then =
dx
(A) cos t2 (B) − sin t2 (C) − sin 2t (D) sin t2 (E) cos 2t
,6. Find the area inside one loop of the curve r = sin 2θ.
π π π π
(A) (B) (C) (D) (E) π
16 8 4 2
1
7. Find the length of the arc of the curve defined by x = t2 and
2
1 3
y = (6t + 9) 2 , from t = 0 to t = 2.
9
(A) 8 (B) 10 (C) 12 (D) 14 (E) 16
8. If f is a vector-valued function defined by f (t) = (sin 2t, sin2 t),
then f 00 (t) =
(A) (−4 sin 2t, 2 cos 2t)
(B) (− sin 2t, − cos2 t)
(C) (4 sin 2t, cos2 t)
(D) (4 sin 2t, −2 cos 2t)
(E) (2 cos 2t, −4 sin 2t)
9. A solid is formed by revolving the region bounded by the x-axis,
the lines x = 0 and x = 1 and the parametric curve x = sin t, y =
1 + cos2 (2t) about the x-axis. Write down the integral which will
compute the volume of the solid.
Z π Z π
2 2
4
(A) π (1 + cos (2t)) dt (C) π (sin2 t + (1 + cos2 (2t))2 ) dt
0 0
Z π
2
(D) 2π (1 + cos2 (2t))2 dt
0
π
π
Z
2 Z
2
(B) π (1 + cos2 (2t))2 dt (E) π cos t(1 + cos2 (2t))2 dt
0 0
, 10. The length of the curve determined by x = 3t and y = 2t2 from
t = 0 to t = 9 is
Z 9p
(A) 9t2 + 4t4 dt
0
Z 162 p
(B) 9 − 16t2 dt
0
Z 162 p
(C) 9 + 16t2 dt
0
Z 3p
(D) 9 − 16t2 dt
0
Z 9p
(E) 9 + 16t2 dt.
0
11. The length of the curve determined by x = 2t3 and y = t3 from
t = 0 to t = 1 is
√
5 5 3 √
(A) (B) (C) (D) 5 (E) 3
7 2 2
12. The area of the region inside
√ the polar curve r = 4 sin θ but out-
side the polar curve r = 2 2 is given by
Z 3π/4
(A) 2 (4 sin2 θ − 1) dθ
π/4
1
Z 3π/4 √ 2
(B) (4 sin θ − 2 2) dθ
2 π/4
1
Z 3π/4 √
(C) (4 sin θ − 2 2) dθ
2 π/4
1 3π/4
Z
(D) (16 sin2 θ − 8) dθ
2 π/4
1 3π/4
Z
(E) (4 sin2 θ − 1) dθ.
2 π/4
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