This document provides examples of using calculus to find the area of different shapes and regions bounded by curves. It includes problems that involve using integration, the midpoint rule, and finding the intersection points of curves graphically in order to approximate areas. Some specific problems calculate the area of triangles, regions bounded by sine and cosine curves, and the region between two curves.

1. 420 |||| CHAPTER 6 APPLICATIONS OF INTEGRATION
,
22. ,
23. , , ,
24. , ,
25.
26.
27. , , ,
28. , , ,
29–30 Use calculus to find the area of the triangle with the given
vertices.
, ,
30. , ,
31–32 Evaluate the integral and interpret it as the area of a
region. Sketch the region.
31.
32.
33–34 Use the Midpoint Rule with to approximate the
area of the region bounded by the given curves.
33. , ,
34. , ,
; 35–38 Use a graph to find approximate -coordinates of the points
of intersection of the given curves. Then find (approximately) the
area of the region bounded by the curves.
35. ,
36.
37. ,
38. , y ෇ x10
y ෇ x cos x
y ෇ x3
Ϫ 3x ϩ 4y ෇ 3x2
Ϫ 2x
y ෇ ex
, y ෇ 2 Ϫ x2
y ෇ x4
y ෇ x sin͑x2
͒
x
x ෇ 0y ෇ xy ෇ s3
16 Ϫ x3
0 ഛ x ഛ 1y ෇ cos2
͑␲x͞4͒y ෇ sin2
͑␲x͞4͒
n ෇ 4
y
4
0 Խsx ϩ 2 Ϫ x Խ dx
y
␲͞2
0 Խsin x Ϫ cos 2x Խdx
͑5, 1͒͑2, Ϫ2͒͑0, 5͒
͑Ϫ1, 6͒͑2, 1͒͑0, 0͒29.
x ജ 04x ϩ y ෇ 4y ෇ 8x2
y ෇ 3x2
x Ͼ 0y ෇
1
4 xy ෇ xy ෇ 1͞x
y ෇ ԽxԽ, y ෇ x2
Ϫ 2
y ෇ x2
, y ෇ 2͑͞x2
ϩ 1͒
0 ഛ x ഛ ␲y ෇ 1 Ϫ cos xy ෇ cos x
x ෇ ␲͞2x ෇ 0y ෇ sin 2xy ෇ cos x
y ෇ xy ෇ sin͑␲x͞2͒
x ෇ y2
Ϫ 1x ෇ 1 Ϫ y2
21.1–4 Find the area of the shaded region.
1. 2.
4.
5–28 Sketch the region enclosed by the given curves. Decide
whether to integrate with respect to x or y. Draw a typical approx-
imating rectangle and label its height and width. Then find the
area of the region.
5.
6.
7. ,
8.
10.
11. ,
12. ,
,
14. , ,
15. , ,
16.
17. , ,
18. , , ,
19. ,
20. , x ෇ y4x ϩ y2
෇ 12
x ෇ 4 ϩ y2
x ෇ 2y2
x ෇ 3x ෇ Ϫ3y ෇ x2
y ෇ 8 Ϫ x2
x ෇ 9y ෇
1
2 xy ෇ sx
y ෇ x3
Ϫ x, y ෇ 3x
Ϫ␲͞3 ഛ x ഛ ␲͞3y ෇ 2 sin xy ෇ tan x
0 ഛ x ഛ 2␲y ෇ 2 Ϫ cos xy ෇ cos x
y ෇ x2
Ϫ 6y ෇ 12 Ϫ x2
13.
y ෇ 4x Ϫ x2
y ෇ x2
y2
෇ xy ෇ x2
y ෇ 1 ϩ sx, y ෇ ͑3 ϩ x͒͞3
y ෇ 1͞x, y ෇ 1͞x2
, x ෇ 29.
y ෇ x2
Ϫ 2x, y ෇ x ϩ 4
y ෇ x2
y ෇ x
y ෇ sin x, y ෇ ex
, x ෇ 0, x ෇ ␲͞2
y ෇ x ϩ 1, y ෇ 9 Ϫ x2
, x ෇ Ϫ1, x ෇ 2
x
y
(_3,€3)
x=2y-¥
x=¥-4y
x
x=¥-2
x=ey
y=1
y=_1
y3.
x=2
y=œ„„„„x+2
y=
1
x+1
x
y
y=x
y=5x-≈
x
y
(4,€4)
EXERCISES6.1