Math 2A - Chapter 12 Test Fall ’07 Name________________________________ Show your work for credit. Write all responses on separate paper. Do not use a calculator.
1. Write each area as an iterated integral. You do not need to compute its value. a. where b. The area of the region inside the circle and outside the circle .
2.
Interpret the integral in polar form and evaluate. 3.
Find the volume of the region E = 4. Suppose a mass density function is distributed over the laminar region bounded by and a. Find the mass of this laminar region. b.
Find the x-coordinate
of its center of mass. 5.
Evaluate . It may help to reverse the order of
integration. 6.
Find the area of the surface z = x + y2 that lies above the triangle
with 7.
Evaluate where E
is the tetrahedron with vertices (0,0,0), (0,1,0), (1,1,0) and (0,1,1). 8.
Use cylindrical coordinates to evaluate where E
is the region that lies inside the cylinder and between the planes z = 5 and z
= 4. 9.
Use spherical coordinates to evaluate ,
where E lies between 10. Use
the given transformation to evaluate ,
where R is the region bounded by
the
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Math 2A - Chapter 12 Test Solutions Fall ’07
1. Write each area as an iterated integral. You do not need to compute its value. a.
where
2.
Interpret the integral in polar form and evaluate. 3.
Find the volume of the region E = 4. Suppose a mass density function is distributed over the laminar region bounded by and a.
Find the mass of this laminar region.
ANS: is a parameterization of the surface, so the area is 7.
Evaluate with E
the tetrahedron having vertices (0,0,0), (0,1,0), (1,1,0) and (0,1,1). 8.
Use cylindrical coordinates to evaluate where E
is the region that lies inside the cylinder and between the planes z = 5 and z
= 4. 9.
Use spherical coordinates to evaluate ,
where E lies between
Corners of a rectangular region
in the uv-plane. Thus the integral is |