Unformatted text preview: Math 2220 Section 5.1 : Problem Set 5 Spring 2010 √ ln x dx dy . 6. Evaluate the integral xy 11 8. Find the volume of the region bounded on the top by the plane z = x + 3y + 1 , on the
9 e bottom by the xy -plane, and on the sides by the planes x = 0 , x = 3 , y = 1 , y = 2 . For the following two problems, calculate the integral and indicate what region in R3 the integral is computing the volume of.
2 3 10.
0 3 1 2 dx dy .
1 0 14.
−2 |x| sin πy dy dx . Section 5.2 : In the following two problems evaluate the integral and sketch the region in the xy -plane determined by the limits of integration.
2 x2 0 1 4.
0 y dy dx . √2
−1 0 3 dx dy . 12. Integrate the function f (x, y ) = 3xy over the region bounded by y = 32x3 and √ y = x. 14. Evaluate y = 3. 16. Evaluate
D D 3y dA where D is the region bounded by xy 2 = 1 , y = x , x = 0 and (x2 + y 2 ) dA where D is the region in the ﬁrst quadrant bounded by y = x , y = 3x , and xy = 3 . 28. (a) Show that if R = [a, b] × [c, d] , f is continuous on [a, b] , and g is continuous on
b d [c, d] , then f (x)g (y ) dA =
R a f (x) dx
c g (y ) dy (b) What can you say about
D f (x)g (y ) dA if D is not a rectangle? Speciﬁcally, what if D is an elementary region of type 1? ...
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