지난문제 게시판읽기 ( 2016학년도 1학기 미적분학및연습1 2차시험 ) | 수학과

Calculus I (Math161)

Exam 2 (Spring, 2016)

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1. (12 pts.) Evaluate the integral

Z _{(x − 2)}2_tan−1_{(2x) − 12x}3_{− 3x}

(4x2_{+ 1)(x − 2)}2 dx

.

2. (13 pts.) Evaluate the integral

Z 1₂ −1 2 1 x −√1 − x2dx. 1

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3. (12 pts.) Determine if the following improper integral converges : Z 1 0 ex_cos πx 2
p x10_{+ sin}−1_xdx

4. (13 pts.) Find all (p, q) such that the improper integral R1

0 x

p _{ln x}
q

dx converges.

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5. (12 pts.) Find the area of the surface generated by rotat-ing the curve y = x₂2, 0 ≤ x ≤ 1, about the x-axis.

6. (13 pts.) Find the area enclosed by one loop of the curve C which is defined by

C : x(t) =1

2(sin t + sin 3t) and y(t) = 1

2(cos t − cos 3t)

for t ∈ (−∞, ∞).

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7. (12 pts.) A curve is defined by the parametric equations

x(t) = Z t 1 cos u u du and y(t) = Z t 1 sin u u du for t ≥ 1.

Find the length of the arc of the curve from the origin to the nearest point where there is a vertical tangent line.

8. (13 pts.) Let n be a positive integer. Find the area of the region that lies inside the curve r = 5 + sin nθ − cos nθ and outside of the circle x2+ y2= 25.