? ? 0 ? cos y 0 x ? ? sin y ? x ? y = ? 1 3 ? 2 0 ? 1 y / 2 e ( x 2 ) ? x ? y =

Question

??0?cosy0x??siny?x?y=?13

?20?1y/2e(x2)?x?y=e?1

(Hint: Reverse the order of integration.)

If you reverse the order of integration, ?0?2?4?x2 ? 0f(x,y)?y?x=?20?0?4?y2 ?f(x,y)?x?y.

??1?2xxye?x?y?x=152e

?10?33yex?x?y=2e3+1

Use a double integral to find the volume of the solid in the first octant bounded above by the plane x + y + z = 4 and below by the rectangle on the xy-plane : {(x, y) : 0?x?1,??0?y?2}.

5

2

4

8

Find the average value of f(x,y)=y over the rectangle with vertices (0,0), (2,0), (2,3), and (0,3).

1/2

2/3

4/3

3/2

Use a double integral to find the volume in the first octant bounded by f(x,y)=1?xy,??x=y2, and x=y.

1/6

1/8

1/12

5/12

Evaluate the double integral by changing to polar coordinates: ?1?1?1?x2 ? 0e(x2+y2)?y?x.

?(e?1)

?(e?1)2

?e

?e2

Evaluate ?Rcos??dAwhere R is the second quadrant region lying inside the circle given by r = 4, and outside the circle given by r = 2.

12

-12

6

-6

Explanation / Answer

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