=========================preview======================
(MATH102)[2007](s)midterm~3885^_82714.pdf
Back to MATH102 Login to download
======================================================
HKUST MATH 102
Third Midterm Examination

Multivariable and Vector Calculus
12 April 2007
Answer all .ve questions Time allowed C 120 minutes

Directions C This is a closed book examination. No talking or whispering are allowed. Work must be shown to receive points. An answer alone is not enough. Please write neatly. Answers which are illegible for the grader cannot be given credit.
Note that you can work on both sides of the paper and do not detach pages from this exam packet or unstaple the packet.
Student Name:
Student Number:

Tutorial Session:

Question No. Marks 1 /20 2 /20 3 /20 4 /20 5 /20 Total /100

22
yy
22
Theintersection ofthetwo surfaces(elliptic cylinders) x+ =1and z+ = 1 consists of
22
two curves.

(a)
Parameterize each curve in the form r(t)=(x(t),y(t),z(t)).

(b)
What is the arc length of one of the curves?

(c)
Find the volume bounded by the two elliptic cylinders above.


2

y
2

(d) Find thesurfaceareaof thepartof theelliptic cylinder z+ =1thatlieswithintheelliptic
2
2
y
2
cylinder x+ =1.
2

82 2 x 2
ye

(a) Evaluate dx dy by reverse the order of integration.
8
1/3 x
0 y

(b) If Q =[.1, 1] [0, 2], evaluate the double integral |y .x2 |dx dy, given that it exists.
Q
(Hint: reverse the order of integration)

22

Find the volume of the solid that lies above the cone z = x+ yand between the sphere
222

x+ y+ z= z.(a)Use spherical coordinates,(b) Use cylindrical coordinates.
(a) If x(u, v)and y(u, v)have continuous .rst partial derivatives, and that .(x, y)
.=0 at (u, v) (one-to-one map).
.(u, v)
.(x, y)1
Prove =
.(u, v) .(u, v)
.(x, y)

(x .y)4

(b) Evaluate dx dy, where D is the triangular region bounded by the x and y axes
(x + y)4
D
and the line x + y =1.

Sketch the region of integration of the following iterated integral


x
sin(xy)dy dx dz.

0 z 0

(a)
Express the above integral as .ve other di.erent iterations of the tirple integrals.

(b)
Then evaluate the integral.