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

434 Chapter 6 Applications of Integration
We must integrate between the appropriate y-values, y − 22 and y − 4. Thus
A − y 4 22
sx R 2 x L d dy − y 4 22 fsy 1 1d 2 ( 1 2 y 2 2 3)g dy
− y 4 (2 1 2 y 2 1 y 1 4) dy
22
y
y= œ„„„„„ 2x+6
(5, 4)
− 2 S D 1 y 3
1 y 2
4
2 3 2 1 4y G22
A
y=x-1
− 2 1 6 s64d 1 8 1 16 2 ( 4 3 1 2 2 8) − 18
n
3 A¡
y=_ œ„„„„„ 2x+6
FIGURE 16
0 x
(_1, _2)
Note We could have found the area in Example 7 by integrating with respect to x
instead of y, but the calculation is much more involved. Because the bottom boundary
consists of two different curves, it would have meant splitting the region in two and
computing the areas labeled A 1 and A 2 in Figure 16. The method we used in Example 7
is much easier.
1–4 Find the area of the shaded region.
10. y − sin x, y − 2xy, x > 0
1. y
2.
y
11. x − 1 2 y 2 , x − y 2 2 1
y=Œ„ x
(1, e)
12. 4x 1 y 2 − 12, x − y
(1, 1)
x=8
0
y=1/x
x
3. y
4.
x=¥-2 y=1
(_3, 3)
y=´
y=xe ≈
0
x
y
x=¥-4y
13–28 Sketch the region enclosed by the given curves and find
its area.
13. y − 12 2 x 2 , y − x 2 2 6
14. y − x 2 , y − 4x 2 x 2
15. y − sec 2 x, y − 8 cos x, 2y3 < x < y3
16. y − cos x, y − 2 2 cos x, 0 < x < 2
x=e y
x
17. x − 2y 2 , x − 4 1 y 2
y=_1
x
18. y − sx 2 1 , x 2 y − 1
x=2y-¥
19. y − cos x, y − 4x 2 2 1
20. x − y 4 , y − s2 2 x , y − 0
5–12 Sketch the region enclosed by the given curves. Decide
whether to integrate with respect to x or y. Draw a typical approximating
rectangle and label its height and width. Then find the area
of the region.
5. y − e x , y − x 2 2 1, x − 21, x − 1
6. y − sin x, y − x, x − y2, x −
7. y − sx 2 2d 2 , y − x
8. y − x 2 2 4x, y − 2x
9. y − 1yx, y − 1yx 2 , x − 2
21. y − tan x, y − 2 sin x, 2y3 < x < y3
22. y − x 3 , y − x
23. y − s 3 2x , y − 1 8 x 2 , 0 < x < 6
24. y − cos x, y − 1 2 cos x, 0 < x <
25. y − x 4 , y − 2 2 | x |
26. y − sinh x, y − e 2x , x − 0, x − 2
27. y − 1yx, y − x, y − 1 4 x, x . 0
28. y − 1 4 x 2 , y − 2x 2 , x 1 y − 3, x > 0
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.