1. The figure to the right shows a polyhedron with five vertices, eight edges, and five faces.
(a) Find the equation for the plane containing the points a
"ß !ß " , !ß "ß " b
, and a
"ß "ß !
H 1,0,1 L
H 1,0,0 L x z
H 1,1,0 L
H 0,1,1 L
H 0,1,0 L y
(b) Sketch the base of this polyhedron on the following axes. Label each of the three vertices, and indicate the equations for each of the three lines.
PRACTICE SPACE y
FINAL ANSWER y x x
(c) Use a double integral to find the volume of this polyhedron. You must show your work to receive full credit.
2. The goal of this problem is to estimate the value of the integral
$ where is the rectangle e a
Bß C ± ! Ÿ B Ÿ "Þ&ß ! Ÿ C Ÿ "Þ& f
(a) Use nine subrectangles and the midpoint rule to estimate the value of this integral. Record the height that you are using for each box on the table below:
Note: If you do this part correctly, your answer should be approximately #Þ&(!) .
(b) Use a spreadsheet such as Microsoft Excel to estimate this integral using 900 rectangles (30 in each direction) and the midpoint rule. Write your final answer below, and attach a printout of your spreadsheet to your homework assignment.
3. Cottage Lake is a small lake in King County, Washington. The following figure shows a
bathymetric chart for the lake, i.e. a contour plot of the depth of the water. Depths are given in feet, and each square has a side length of 500 feet:
Using this contour plot, estimate the total volume of water in Cottage Lake as accurately as you can.
(Show your work, and express your final answer in cubic feet. Your answer must be correct to within 20% to receive full credit.)