010.104-2006-2-fin

# 010.104-2006-2-fin - Honor Calculus2 Final Exam 2006 12 9...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Honor Calculus2 Final Exam 2006 12 9{ 13r 15r 4 Z 9 < V: s2: M K+ c a Z @\ %V IU lǣ k k( 120). \ 1. (a) (7 pts) Let c(t) = (sin2 t2 , cos2 t2 , cos(2t2 )) (0 t 2 c (x + 3yz)ds. 2) be a path in space. Evaluate (b) (8 pts) Let C1 be a straight line from (0, 0) to (1, 1) and C2 be a portion of a circle x2 + y 2 = 2, x y from (1, 1) to (-1, -1). For F(x, y) = x2 i + (1 + y 2 )j, evaluate C1 +C2 F ds. 2. (10 pts) Evaluate the line integral (2xex + arctan2 y)dx + (sin3 y cos y + etan x )dy, C 2 where C is the boundary of the rectangle [0, ] [0, 1], oriented in the counterclockwise 4 direction. 3. (15 pts) The solid angle of the region R at the point P means the area of the projection of R into unit sphere centered at P . For example, the solid angle of the first octant 1 x > 0, y > 0, z > 0 at the origin is 8 4 = . 2 Let C be the cone given by z x2 + y 2 . Find the solid angle of C at the vertex of C. 4. (15 pts) Let S be a cylinder given by x2 + y 2 = 4 and 0 z 1. And let F(x, y, z) = (xy)i+(y+x sinh z)j+(y sinh z)k. (Note that sinh z = 1 (ez -e-z ).) Evaluate S curl FdS. 2 5. (15 pts) Let S be the union of S1 and S2 , where S1 = {(x, y, z)|z = x2 + y 2 , 1 z 2} and S2 = {(x, y, z)|x2 + y 2 + (z - 2)2 = 2, z 2}. For F = (zx + z 2 y + x)i + (z 3 yx + y 2 )j + z 4 x2 k, compute S ( F) dS. 6. (15 pts) Let W be a region x2 + y 2 1, 0 z 1 in space and W be the boundary of W with outward normal orientation. Evaluate W F dS for F(x, y, z) = (xy 2 + yz)i + (xz + yz)j + (x2 z + z 2 )k. 7. (a) (10 pts) Let F = (2xyez +6x)i+(x2 ez +z)j+(x2 yez +y)k. Prove that F is irrotational and find a function f such that f = F. (b) (10 pts) Let F = xi + yj - 2zk. Find G such that curl G = F. x 8. (15 pts) Let F = x2-y 2 i+ x2 +y2 j. For the closed curve C in Figure 1 with counter-clockwise +y orientation, compute C F ds. Figure 1: Figure for Problem 8 ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online