Math 215 Study Guide for Midterm 1 (Sections 15.1 15.7)
Review Section 15.8 from notes and homework
1. Estimate the volume of the solid that lies above the square R [0,2] [0,2] and below
the elliptic paraboloid z 16 x 2 2 y 2 . Use a Riemann sum with m n
Math 215  Final Study Guide Winter 2015
Chapter 15:
1. Find the mass and center of mass of the triangular region with vertices
(0,0), 2,1, and (0,3) if the density function is ( x, y ) x y.
2. Use a triple integral to find the volume of the tetrahedron T
Chapter 13
Monday, April 24, 2017
9:39 AM
13.1  Double Integrals over Rectangular Regions
13.2  Double Integrals over General Regions
Type 1, bounded above and below
Type 2, bounded left and right

13.3  Double Integrals in Polar Coordinates
13.4 
Chapter 14
Wednesday, April 26, 2017
9:01 AM
14.1  Vector Fields
 Radial vector fields:
MAT 215 Page 1
14.1
Monday, April 24, 2017
9:32 AM
Vector Field
Roughly speaking, a vector field is a field of vectors which follow a certain rule (pattern) given by
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Math 215
Sample Questions Test 2
Sketch the vector field F.
1.
F ( x, y ) = x, y
2.
F ( x, y ) = y i + j
4.
f ( x, y, z ) = xy 2 yz 3
Find the gradient vector field of f.
3.
f ( x, y ) = x 5 4 x 2 y 3
Determine whether or not F is a conservative vector fi
Math 215
Practice Test 1
Evaluate the following multiple integrals
1.
35
2
sin
2
2
x cos y dy dx
2.
00
00
1
3.
1
2
e
0y
xy dx dy
x2
ln 10 10
dx dy
4.
1
ln y dy dx
0 ex
2
(Hint: try changing the order of integration)
111
5.
x
2
2
2
y z dx dy dz
6.
1 1
Math 215 Study Guide for Midterm 2 (Sections 15.8 16.4)
1. Use spherical coordinates to evaluate
zdV , where E lies between the spheres
E
2
2
2
2
2
2
x y z 1 and x y z 4 in the first octant.
2. Find the volume of the solid that lies within the sphere x 2