Math 211: Worksheet 15.2 Day 30
Calculate the following double integrals. Sketch the region that the integral is over.
(1) Z2
1Z4
0
1
x+ydydx
(2) Z ZD
x2+y2dA where Dis the area enclosed by y= 2x2and y= 1 + x2
(3) Z ZD
ex+ydA where Dis given by 2 ≤x≤4 and x+ 1 ≤y≤12 −x
(4) Z ZD
P(x, y)dA where P(x, y) is the plane containing the points (a, 0,0),(0, a, 0), and
(0,0, a). The region Dis between the x-axis, y-axis, and the intersection of the plane with
the xy-plane
Recall the Cobb-Douglas model for production. The production and cost of production of goods is
modeled by the functions
P=bxαy1−α,
C=mx +ny
•Pis units produced,
•Cis total cost,
•xis amount of labor used,
•yis amount of capital investment,
•mis the cost of a unit of labor,
•nis the cost of a unit of capital,
•band αare positive constants and α≤1.
For this problem assume that the cost of labor mand the cost of a unit of capital nare fixed. We
also assume that m, n, x, y ≥0.
(1) Calculate the average production if the total budget varies between c1and c2, you may
assume c1< c2.
(2) Calculate the average cost if the production varies between p1and p2, you may assume that
p1< p2. For this part you should also assume that x, y ≥1 (without this assumption we
run into a region with “infinite area” to integrate over).
(Hint: This integral needs to be split up into two regions.)
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