Fig. 44-28
44.46
Evaluate
44.45
The region of integration (Fig. 44-29) consists of all points in the first quadrant above the circle x
2 + y
2 = 1
and under the line y = x. Transform to polar coordinates, noting that x = 1 is equivalent to r = sec 0.
If the depth of water provided by a water sprinkler in a given unit of time is 2 ' feet at a distance of r feet from the
sprinkler, find the total volume of water within a distance of a feet from the sprinkler after one unit of time.
44.43
44.44
Fig. 44-26
Fig. 44-27
Use polar coordinates to evaluate
The region of integration is indicated in Fig. 44-27.
Use polar coordinates to evaluate
y = V2x - x
2
is equivalent to y
2 = 2x - x
2 , y >0, or the upper half of the circle (x - I)
2 + y =1
(see Fig. 44-28). In polar coordinates, x
2 + y
2 = 2x is equivalent to r
2 = 2rcos0, or r = 2 cos 6. Thus,
Problem 44.29).
(by
MULTIPLE INTEGRALS AND THEIR APPLICATIONS
413
44.42
Find the average distance from points in the unit disk to a fixed point on the boundary.
For the unit circle r = 2sin0, with the pole as the fixed point (Fig. 44-26), the distance of an interior
point to the pole is r.
Thus,