y = 2		( y ³ 2).
	ì						3
	ïx = 24cos							t,
	í
	ïy = 2sin3 t,
15.25. î						(x ³ 9				).
	x = 9
			3						3
	ìx = 2	(t - sint),
	ï
15.27.	íy = 2	(1- cost ),
	ï
	î
	y = 2 (0 < x < 4π , y ³ 2).
	ì
				2 cost,
	ïx = 2
	í
				2 sint,
15.29. ïy = 5
	î		( y ³ 5).
	y = 5
	ì						3
	ïx = 32cos							t,
	í
	ìx = 9cost,
	í
15.23. îy = 4sint,

15.31. ïîy = 3sin3 t,

x =123 (x ³123).

	ìx = 8(t - sin t),
	ï
15.24.	íy = 8	(1- cost ),
	ï
	î
	y =12 (0 < x <16π , y ³12).
	ìx = 3cost,
	í
15.26. îy = 8sint,
	y = 4						(y ³ 4				).
				3						3
	ì							3	t,
				2 cos
	ïx = 4
	í
			2 sin3 t,
15.28. ïy =
	î		(x ³ 2).
	x = 2
	ìx = 4	(t - sint),
	ï
15.30.	íy = 4	(1- cost ),
	ï
	î

y = 6 (0 < x < 8π , y ³ 6).

	Задача 16. Вычислить площади фигур, ограниченных линиями, заданными
полярных координатах.
16.1. r = 4cos3ϕ,				r = 2 (r ³ 2).			16.2. r = cos2ϕ.
	r =			r = sinϕ,
16.3.		3cosϕ,					16.4. r = 4sin3ϕ, r = 2 (r ³ 2).
	(0 £			ϕ £ π 2).
	r = 2cosϕ,			r = 2
16.5.					3sinϕ,		16.6. r = sin3ϕ.
	(0 £			ϕ £ π 2).

96

16.7. r = 6sin3ϕ,					r = 3 (r ³ 3).	16.8. r = cos3ϕ.
r = cosϕ,						r = sinϕ,
16.9. r =		sin(ϕ -π 4),				16.10. r =		cos(ϕ -π 4),
	2						2
(-π 4 £ ϕ £ π 2).						(0 £ ϕ £ 3π 4).
16.11. r = 6cos3ϕ, r = 3 (r ³ 3).						16.12. r =1 2 + sinϕ.
						r =		cos(ϕ -π 4),
r = cosϕ,					r = sinϕ,		2
16.13. (0 £ ϕ £ π 2).						16.14. r =	2 sin (ϕ -π 4),
						(π 4 £ ϕ £ 3π 4).
16.15. r = cosϕ,					r = 2cosϕ.	16.16. r = sinϕ,			r = 2sinϕ.
16.17. r =1+				cosϕ.		16.18. r =1 2 + cosϕ.
			2
16.19. r =1+				sinϕ.		16.20. r = (5 2)sinϕ, r = (3 2)sinϕ.
			2
16.21. r = (3 2)cosϕ, r = (5 2)cosϕ.						16.22. r = 4cos4ϕ.
16.23. r = sin6ϕ.						16.24. r = 2cosϕ,			r = 3cosϕ.
16.25. r = cosϕ + sinϕ.						16.26. r = 2sin 4ϕ.
16.27. r = 2cos6ϕ.						16.28. r = cosϕ − sinϕ.
16.29. r = 3sinϕ,					r = 5sinϕ.	16.30. r = 2sinϕ,			r = 4sinϕ.
16.31. r = 6sinϕ,					r = 4sinϕ.

Задача 17. Вычислить длины дуг кривых, заданных уравнениями в прямоугольной системе координат.

							x2		ln x
	17.1. y = ln x, 3 £ x £ 15.					17.2. y =		-		, 1	£ x £ 2.
							4		2

97

17.3. y =

0 ≤ x ≤ 7 9.

17.3. y = ln

5

,

≤ x ≤

1− x2

+ arcsin x,

3

8.

2x

17.5. y = −ln cos x,

0 ≤ x ≤ π 6.

17.6. y = ex + 6,

ln

≤ x ≤ ln

8

15.

17.7. y = 2 + arcsin

+

1 4 ≤ x ≤1.

x − x2

,

x

17.8. y = ln(x2 −1),

2 ≤ x ≤ 3.

17.9. y =

0 ≤ x ≤ 8 9.

17.10. y = ln(1− x2 ),

0 ≤ x ≤1 4.

1− x2

+ arccos x,

17.11. y = 2 + ch x,

0 ≤ x ≤ 1.

17.12. y =1− ln cos x,

0 ≤ x ≤ π 6.

17.13. y = ex +13,

ln

≤ x ≤ ln

15

24.

17.14. y = −arccos

+

0 ≤ x ≤ 1 4.

x − x2

,

x

17.15. y = 2 − ex ,

ln

≤ x ≤ ln

3

8.

17.16. y = arcsin x −

1− x2

,

0 ≤ x ≤ 15 16.

17.17. y =1− lnsin x,

π 3 ≤ x ≤ π 2.

17.18. y =1− ln (x2 −1), 3 ≤ x ≤ 4.

17.19. y =

− arccos

+ 5,

1 9 ≤ x ≤ 1.

x − x2

x

17.20. y = −arccos x +

0 ≤ x ≤ 9 16.

1− x2

+1,

17.21. y = lnsin x,

π 3 ≤ x ≤ π 2.

17.22. y = ln7 − ln x,

≤ x ≤

3

8.

17.23. y = ch x + 3,

0 ≤ x ≤ 1.

17.24. y = 1+ arcsin x −

0 ≤ x ≤ 3 4.

1− x2

,

17.25. y = lncos x + 2,

0 ≤ x ≤ π 6.

x

ex + e−x

17.26. y = e

+ 26,

8 ≤ x ≤ ln

ln

24.

17.27. y =

+ 3,

0 ≤ x ≤ 2.

2

17.28. y = arccos

−

0 ≤ x ≤ 1 2.

x − x2

+ 4,

x

ex + e− x + 3

x

17.29. y =

,

0 ≤ x ≤ 2.

17.30. y = e

+ e,

ln

3 ≤ x ≤ ln

15.

4

17.31. y =

1− ex − e−x

,

0 ≤ x ≤ 3.

2

98

Задача 18. Вычислить длины дуг кривых, заданных параметрическими уравнениями.

ìx = 5(t - sint ),

ï

18.1. íïîy = 5(1- cost), 0 £ t £ π.

ìx = 4(cost + t sint),

ï

18.3. íïîy = 4(sint - t cost), 0 £ t £ 2π.

ìïx =10cos3 t,

í

18.5. ïîy =10sin3 t, 0 £ t £ π2.

ìx = 3(t - sint ),

ï

18.7. íïîy = 3(1- cost),

π £ t £ 2π.

ìx = 3(cost + t sin t),

ï

18.9. íïîy = 3(sin t - t cost), 0 £ t £ π3.

ìïx = 6cos3 t,

í

18.11. ïîy = 6sin3 t,

0 £ t £ π3.

ìïx = 2,5(t - sint),

18.13.íïîy = 2,5(1- cost ),

π2 £ t £ π.

ìx = 3(2cost - cos2t),

ï

18.2. íïîy = 3(2sint - sin 2t ), 0 £ t £ 2π.

ìïx = (t2 - 2)sint + 2t cost,

18.4.íïy = (2 - t2 )cost + 2t sin t,

î

	ì						0 £ t £ π.
		t	(cost + sint ),
	ïx = e
18.6.	íy = et (cost - sin t),
	ï
	î
					0 £ t £ π.
	ìx =	1 cost -				1 cos2t,
	ï	2				4
18.8.	í	1				1
	ï		sint -				sin 2t,
	ïy =	2				4
	î
		π 2 £ t £ 2π 3.
	ì			2	- 2)sint + 2t cost,
	ïx = (t

18.10.íïy = (2 - t2 )cost + 2t sin t,

î

0 £ t £ π3.

ìx = et (cost + sint ),

ï

18.12. íïîy = et (cost - sin t),

π2 £ t £ π.

ìx = 3,5(2cost - cos2t ),

ï

18.14. íïîy = 3,5(2sint - sin 2t), 0 £ t £ π2.

99

ìx = 6(cost + t sint),

ï

18.15. íïîy = 6(sint - t cost), 0 £ t £ π.

ìïx = 8cos3 t,

í

18.17. ïîy = 8sin3 t,

0 £ t £ π6.

ìx = 4(t - sint ),

ï

18.19. íïîy = 4(1- cost),

π2 £ t £ 2π3.

ìx = 8(cost + t sint),

ï

18.21. íïîy = 8(sin t - t cost), 0 £ t £ π4.

ìïx = 4cos3 t,

í

18.23. ïîy = 4sin3 t,

π6 £ t £ π4.

ìx = 2(t - sint ),

ï

18.25. íïîy = 2(1- cost),

0£ t £ π2.

ìx = 2(cost + t sint),

ï

18.27.íïîy = 2(sint - t cost),

0 £ t £ π2.

ìïx = 2cos3 t,

í

18.29. ïîy = 2sin3 t,

0 £ t £ π4.

ìïx = (t2 - 2)sint + 2t cost,

18.16.íïy = (2 - t2 )cost + 2t sin t,

î

0 £ t £ π2.

ìx = et (cost + sint ),

ï

18.18. íïîy = et (cost - sin t), 0 £ t £ 2π.

ìx = 2(2cost - cos2t),

ï

18.20. íïîy = 2(2sint - sin 2t ), 0 £ t £ π3.

ìïx = (t2 - 2)sint + 2t cost,

18.22.íïy = (2 - t2 )cost + 2t sin t,

î

0 £ t £ 2π.

ìx = et (cost + sint ),

ï

18.24. íïîy = et (cost - sin t), 0 £ t £ 3π2.

ìx = 4(2cost - cos2t),

ï

18.26. íïîy = 4(2sint - sin 2t ), 0 £ t £ π.

ìïx = (t2 - 2)sint + 2t cost,

18.28.íïy = (2 - t2 )cost + 2t sin t,

î

0 £ t £ 3π.

ìx = et (cost + sint ),

ï

18.30. íïîy = et (cost - sin t), π6 £ t £ π4.

100