지난문제 게시판읽기 ( 2011 - 2 미적분학및연습 Ⅱ 2차 시험 ) | 수학과

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적분학 및 연습 I (MATH162)

제 2 차시험 (2011 년 2 학기)

학과 : 학번 : 이름 :

1. (14 점) Use Taylar’s formula to find a quadratic approx-imation of ex_{sin y at the origin. Estimate the error in the}

approximation if |x| ≤ 0.1 and |y| ≤ 0.1.

2. (16 점) Estimate the integrals.

(1) R1 0 Ry 0(x 2_{+ y}2_{)dxdy +}R2 1 R2−y 0 (x 2_{+ y}2_)dxdy (2) R2 0(tan −1_{πx − tan}−1_x)dx 1

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3. (14 점) EvaluateR

CF · dr, where F = xyi + yj − yzk and

r is the curve of intersection of the cylinder x = y2 _{and the}

plane z = y from (0,0,0) to (1,1,1)

4. (14 점) A swimming pool is circular with a 40m diameter. The depth is constant along east-west lines and increases linearly from 2m at the south end to 7m at the north end. Find the volume of water in the pool.

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5. (14 점) Use cylindrical coordinates to fnd the volume of the solid bounded above by the sphere x2_{+ y}2_{+ z}2_{= a}2_and

below by the cone z =px2_{+ y}2_{cotφ where 0 < φ < π/2.}

6. (14 점) Find the volume of the region bounded by the surface√x +py

2+

√

2z = 1 and the coordinate planes.

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7. (14 점) FindRRR

E 1

(x2_+y2_+z2₎n/2dV , where E is the region

bounded by the spheres with center the origin and radii r and R, 0 < r < R.