Math 251 Week in Review Sections 13.1-13.3
1. 13.1 Find the Riemann sum for f ( x , y )  3 xy  2 x
2
1  x  3, 0  y  1 Partition the
region into 4 equal rectangles and use the upper right corner of each as an evaluation point.
2. 13.2 Evaluate each iterated integral over the given rectangle.
42
a)
12
  xe
xy
dydx
  xye
b)
11
dydx
  x sin(
c)
y

2
0 01 x y
2
xy ) dxdy
0 4
00
21
d)
1 2
2
x y
dydx
3. 13.3 Sketch the region and set up

f ( x , y ) dA in two orders and evaluate it.
R
2
a)
f ( x, y )  x y
b)
f ( x , y )  xe
c)
f ( x, y )  x
3
y
R is the triangle
R is bounded
R is bounded
with veti ces (0,0), (2,4), (0,4).
by y  x
x  y
2
2
and
y  x  2.
and x  2y  3 .
4. Sketch the region and reverse the order of integration.
1 2x3
1 x
a)

0 x
f ( x , y ) dydx
b)
3
 
0
x
f ( x , y ) dydx
2
1 1
5. Evaluate
 x
3
3
sin y dydx
0 x2
6. Find the volume of the region in the first octant between the planes z  2 x  2 y and
x  y  z  6 . Sketch the solid and its projection in the x,y-plane.
2
2
7. Find the volume of the region inside the paraboloid z  9  x  y over the triangle with vertices
(0,0,0), (3,0,0) and (0,3,0).
2
8. Set up only., the integral to find the volume of the region between the paraboloids z  x  y
2
2
and z  36  3 x  3 y .
2