James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 8.3 Applications to Physics and Engineering 567
23–24 The masses m i are located at the points P i. Find the
moments M x and M y and the center of mass of the system.
23. m 1 − 4, m 2 − 2, m 3 − 4;
P 1s2, 23d, P 2s23, 1d, P 3s3, 5d
24. m 1 − 5, m 2 − 4, m 3 − 3, m 4 − 6;
P 1s24, 2d, P 2s0, 5d, P 3s3, 2d, P 4s1, 22d
25–28 Sketch the region bounded by the curves, and visually estimate
the location of the centroid. Then find the exact coordi nates
of the centroid.
25. y − 2x, y − 0, x − 1
26. y − sx , y − 0, x − 4
27. y − e x , y − 0, x − 0, x − 1
28. y − sin x, y − 0, 0 < x <
;
37. Find the centroid of the region bounded by the curves
y − x 3 2 x and y − x 2 2 1. Sketch the region and plot the
centroid to see if your answer is reasonable.
38. Use a graph to find approximate x-coordinates of the points
of intersection of the curves y − e x and y − 2 2 x 2 . Then
find (approximately) the centroid of the region bounded by
these curves.
39. Prove that the centroid of any triangle is located at the point
of intersection of the medians. [Hints: Place the axes so that
the vertices are sa, 0d, s0, bd, and sc, 0d. Recall that a median
is a line segment from a vertex to the midpoint of the opposite
side. Recall also that the medians intersect at a point
two-thirds of the way from each vertex (along the median)
to the opposite side.]
40–41 Find the centroid of the region shown, not by integration,
but by locating the centroids of the rectangles and triangles (from
Exercise 39) and using additivity of moments.
29–33 Find the centroid of the region bounded by the given
curves.
29. y − x 2 , x − y 2
30. y − 2 2 x 2 , y − x
40.
y
2
1
41.
y
2
1
31. y − sin x, y − cos x, x − 0, x − y4
32. y − x 3 , x 1 y − 2, y − 0
33. x 1 y − 2, x − y 2
_1 0 1 2
x
_2
_1 0 1
_1
2
x
34–35 Calculate the moments M x and M y and the center of mass
of a lamina with the given density and shape.
34. − 4 35. − 6
42. A rectangle 5 with sides a and b is divided into two parts
5 1 and 5 2 by an arc of a parabola that has its vertex at one
corner of 5 and passes through the opposite corner. Find the
centroids of both 5 1 and 5 2.
y
y
y
3
0 2
x
4
0
_2
4
x
5
0 a
¡
b
x
36. Use Simpson’s Rule to estimate the centroid of the region
shown.
y
4
43. If x is the x-coordinate of the centroid of the region that lies
under the graph of a continuous function f, where a < x < b,
show that
y b
a
scx 1 dd f sxd dx − scx 1 ddy b
f sxd dx
a
2
0 2 4 6 8
x
44–46 Use the Theorem of Pappus to find the volume of the
given solid.
44. A sphere of radius r (Use Example 4.)
45. A cone with height h and base radius r
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