Mathematics 55
2nd Semester 2010-2011 Exercise Set II
1. Determine the volume of the solid bounded by the planes x = 0, y = 0, x + y + z = 1 and 2x + 2y − z = 2. Z 1 Z √1−x2 Z 1−x2 −y2 2. Use cylindrical coordinates to evaluate (1 + x2 + y 2 ) dz dy dx. 0
0
0
r
y2 x2 + and above by x2 + y 2 + z 2 = 2. Set-up an iterated 3 3 triple integral in spherical coordinates that gives the mass of G if the density at a point P (x, y, z) is the distance of P from the xy-plane. ~ (x, y, z) = hxey , y, zey i. 4. Let F 3. Let G be the solid in the first octant bounded below by z =
~ and curlF ~. (a) Compute divF ~ in moving an object from the origin to the point (2, 4, −2) along the curve defined by (b) Find the by the force F
work done ~ R(t) = t, t2 , −t . ~ (x, y) = ln y − y 2 cos x, 1 − 2y sin x + x . 5. Consider the vector field F y ~ is conservative and find all its potential functions. (a) Show that F R ~ · dR ~ along any path C from (0, 2) to π , 1 . (b) Evaluate C F 2 6. Suppose C consists of the line segment from (0, 1) to (−1, 0), the line segment from (−1, 0) to (1, 0), and the portion of x = 1 − y 2 from (1, 0) to (0, 1). Use the Green’s Theorem to evaluate I tan−1 x − 4xy dx + 3x2 + ey dy . C
Z Z 7. Evaluate the surface integral
2
2
x +y −z √ dS , where S is the portion of z = −4 + x2 + y 2 that is under the xy-plane. 17 + 4z
S
~ (x, y, z) = cos2 x, sin2 x, −z across the portion of the positively oriented plane 2x + 2y + z = 2 in the 8. Determine the flux of F first octant.
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