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(MATH100)[2008](f)final~ma_ysxad^_10409.pdf
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Final Exam of Math100 (All Lecture Sections, Fall 2008)

Lecture section:

Full Name: Student ID:

1.
Evaluate the double integral by converting to polar coordinates. (10 points)





Solution: The double integral is over the right half of the region enclosed by the circle . Consequently,

, which can be evaluated as
.


2.
Use a triple integral to find the volume of the solid in the first octant bounded by the three coordinate planes (0x=, 0y=, and 0z=) and the plane . Here a, b, and c are positive constants. (15 points)





Solution: The solid can be regarded as a simple xy solid and the corresponding triple integral is , which can be evaluated as

.


3.
Find the work done by the force field 23 on a particle moving from (0,0) to (1,1) along the curve 3yx=. (15 points)





Solution: We use
as the parameter t such that the curve is parametrized as
and
. The work is given by the line integral
with
, (0). It can be evaluated as


.


4.
Find the surface area of the portion of the paraboloid 22 that lies below the plane 5=z. (15 points)





Solution: First, we need to find the region for integration. It is observed that the plane intersects the paraboloid at a circle that has radius 2 and lies on the cutting plane, centered at the point Let . Hence the surface area of the desired portion is given by





Since the region
of the double integral above is circular and the integrand contains the term , we may evaluate the last double integral in polar coordinates as follows: Surface area

.


5.
Show that the vector field kex is conservative. Find a potential function such that F.=. Evaluate the line integral where C is the oriented line segment from )0,0,1( to )1,,2(.. (15 points)






Solution: We need to find a potential function
with
. Integrating the first with respect to yields which leads to
, from which an integration with respect to y yields ky
. Differentiating with respect to yields
. It follows that and constant. Choosing the constant to be zero, we obtain a potential function
. Since the given vector field is conservative, the line integral is independent of path, given by
.


6.
Use Stokes Theorem to evaluate the flux where ky and is the portion of the paraboloid 22 above the plane 1=z with upward orientation. (15 points)






Solution:
in which Greens has been used.


7.
Use the Divergence theorem to calculate the surface integral ., where and is the outward pointing surface of the solid bounded by the elliptic cylinder 1942 and the planes 0=z and 2=z. (15 points)







Solution:

Hints: For double integral over region R enclosed by
, we have the following relations:

,

,

.
With and , the answer is



.