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Math 103

Test 3 -- Stage 2

Due November 23, 1999


Instructions


1.
(15 points)
(a)
Sketch the region of integration in the xy-plane for the integral

\begin{displaymath}
\displaystyle \int_{1}^{2} \int_{0}^{x^{3}} e^{y/x} \,dy 
\,dx.\end{displaymath}

(b)
Evaluate the integral above.

2.
(20 points)
(a)
Sketch the region of integration in the xy-plane for the integral

\begin{displaymath}
\displaystyle \int_{0}^{1} \int_{\arctan(y)}^{\pi/4} \sec x \,dx 
\,dy.\end{displaymath}

(b)
Reverse the order of integration and evaluate the integral above.

3.
(20 points) Consider the solid lying above the plane z=0 and inside both the sphere x2+y2+z2=16 and the cylinder x2+(y-2)2=4.
(a)
Sketch the cross-section of this solid in the xy-plane.
(b)
Set up a double iterated integral in polar coordinates to find the volume of this solid. (Do not evaluate.)

4.
(20 points) In 1994, the Discovery Channel sponsored a scientific expedition to study Loch Ness. One of the things that the scientists studied was the amount of biological matter (plankton, fish, etc.) in the loch. Their objective was to decide whether there was enough food available in the loch to support a population of large predators.
(a)
Based on satellite photographs and sonar readings, the scientists are able to approximately describe Loch Ness as the region bounded by the equations z=0, y=x2, x=y2, and x+y+z=0, with the restriction z<0. Sketch top and side views of Loch Ness based on this description.
(b)
Suppose that the scientists were able to come up with a function $\delta(x,y,z)$ that gave the plankton density at location (x,y) on the surface of the loch and vertical location z in the loch. Using the description above, set up an iterated integral that would give the total amount of plankton in the loch.

5.
(20 points) Consider the solid inside the surface 3z2=x2+y2 and between the planes z=0 and z=4, with density $\delta(x,y,z)=17$.
(a)
Sketch top and side views of the solid.
(b)
Set up iterated integrals for the mass and center of mass of the solid, using either cylindrical or spherical coordinates. For this problem, do not use symmetry. (Do not evaluate. You may use the arctangent function in your limits.)

6.
THIS IS AN EXTRA CREDIT PROBLEM WORTH AN ADDITIONAL 5 POINTS.

Consider the part of the ellipsoid

\begin{displaymath}
x^{2}+y^{2}+\frac{z^{2}}{9}=\frac{10}{9}\end{displaymath}

lying above the plane z=1.
(a)
Sketch the region in the xy-plane over which this surface lies.
(b)
Set up an iterated integral for the surface area of this surface. You may use any formula for surface area. If the formula you use has a cross product in it, do out the cross product. (Do not evaluate the integral.)

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Up: Math 103 Home Page
Joshua Holden
11/21/1999