Name: Calculus III
Exam 3
6 December 2000



Engage! - Captain Jean-Luc Picard


Instructions: Show all work to receive full or partial credit. You may use a scientific calculator/graphing calculator. All University rules and guidelines for student conduct are applicable.

1.

[20 pts] Calculate

$$\iint_S (x^2+y) dA$$

where S is the region between the line y=4x and the curve y=x².

2.

[20 pts] Find the surface area of the plane

x + 2y + 3z = 10

over the disk

$$x^2 + y^2 \leq 1.$$

3.

[20 pts] Find the mass of the object sketched below if $\rho = 3-z$.

$\parbox{1.5in}{\epsfxsize=1.5in \epsfbox{pyramid.eps}}$

4.

[20 pts] Consider a spherical mass of radius a with density

$$\sigma = a^3-(x^2 + y^2)z.$$

(a) Set up but do not evaluate the first moment M_xy in spherical coordinates.















(b) Set up but do not evaluate the first moment M_xy in cylindrical coordinates.















(a) Set up but do not evaluate the first moment M_xy in rectangular coordinates.

5.

[20 pts] If

$$\v F(x,y) = \langle x,\frac{1}{y} \rangle,$$

find $\int_C \v F \cdot d\v r$ where C is the curve below

$\parbox{3in}{\epsfxsize=3in \epsfbox{path.eps}}$