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Национальный Технический Университет Харьковский Политехнический Институт

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<<< < Предыдущая 1 23 / 33

9.11.y =(x +1)2 , y2 = x +1.


19.13.	y = x 36 − x2 , y = 0,
	(0 ≤ x ≤ 6).

9.15.	y = arctg x,						y = 0,
	x =
	x =		3.
	x =						x = 0,
9.17.	x =		ey −1,				x = 0,
9.17.	y = ln 2.
	y = ln 2.
	y =			x		,	y = 0,
9.19.
		1+			x
		1+			x

x=1.

9.21.x =(y −2)3 ,

x= 4y −8.

	y =					x		,		y = 0,

			(	x	2	)	2
9.23.					2	+1
						+1

	x =1.
9.25. x =						1			,	x = 0,
		y
		y				1+ln y
	y =1,					y = e3 .

y = x2 16 − x2 , y = 0,

9.27.(0 ≤ x ≤ 4).

9.29.y =(x −1)2 , y2 = x −1.

9.31.x = 4 −(y −1)2 , x = y2 −4y +3.

9.12.y = 2x − x2 +3, y = x2 −4x +3.

9.14.

x = arccos y,

x = 0,

y = 0.

9.16.

y = x2 8 − x2 , y = 0,

(0 ≤ x ≤ 2

9.18.

y = x 4 − x2 , y = 0,

(0 ≤ x ≤ 2).

y =

y = 0,

9.20.

1+cos x

x =π 2,

x = −π 2.

9.22.

y = cos5 xsin 2x, y = 0,

(0 ≤ x ≤π 2).

9.24.

x = 4 − y2 ,

x = y2 −2y.

9.26.

y = e1 x

y = 0,

x = 2,

x =1.

x = 0,

9.28.

x = 4 − y2 ,

y = 0,

y =1.

9.30.

y = x2 cos x,

y = 0,

(0 ≤ x ≤π 2).

Задача 10. Вычислить длины дуг кривых, заданных уравнениями в прямоугольной системе координат.

10.1. y = ln x,

≤ x ≤

15.

10.2. y =

−

ln x

1 ≤ x ≤ 2.

10.3. y =

0 ≤ x ≤ 7 9.

1− x2 +arcsin x,

10.3. y = ln

≤ x ≤

10.5. y = −lncos x,

0 ≤ x ≤π 6.

10.6. y = ex +6, ln8 ≤ x ≤ ln 15.

10.7. y = 2 +arcsin x +x − x2 , 14 ≤ x ≤1.

10.8. y = ln	(	x2	)
			−1 , 2 ≤ x ≤3.

10.9. y = 1− x2 +arccos x, 0 ≤ x ≤89.

(

)

0 ≤ x ≤1 4.

10.10. y = ln 1− x2

10.11. y = 2 +ch x,

0 ≤ x ≤1.

10.12. y =1−lncos x,

0 ≤ x ≤π 6.

10.13. y = ex +13,

≤ x ≤ ln

24.

10.14. y = −arccos

x − x2 ,

0 ≤ x ≤1 4.

10.15. y = 2 −ex ,

≤ x ≤ ln

10.16. y = arcsin x −

0 ≤ x ≤15 16.

1− x2 ,

10.17. y =1−lnsin x,

)

π 3 ≤ x ≤π 2.

10.18. y =1

−ln

(

3 ≤ x ≤ 4.

−1 ,

10.19. y =

x − x2 −arccos

+5,

1 9 ≤ x ≤1.

10.20. y = −arccos x +

1− x2 +1,

0 ≤ x ≤9 16.

10.21. y = lnsin x,

π 3 ≤ x ≤π 2.

10.22. y = ln7 −ln x,

≤ x ≤

10.23. y = ch x +3,

0 ≤ x ≤1.

10.24. y =1+arcsin x −

0 ≤ x ≤3 4.

1− x2 ,

10.25. y = lncos x +2,

0 ≤ x ≤π 6.

10.26. y = ex +26,

≤ x ≤ ln

24.

10.27. y =

ex +e−x

+3,

0 ≤ x ≤ 2.

10.28. y = arccos

−

x − x2 +4, 0 ≤ x ≤1 2.

10.29. y =

ex +e−x +3

0 ≤ x ≤ 2.

10.30. y = ex +e,

≤ x ≤ ln

15.

10.31. y =

1−ex −e−x

0 ≤ x ≤3.

Задача 11. Вычислить длины дуг кривых, заданных параметрическими

уравнениями.

x =5

(t −sint),

x =3(2cost −cos2t),

11.1. y =5(1−cost),

11.2. y =3(2sint −sin 2t),

0 ≤t ≤π.

0 ≤t ≤ 2π.

x = 4(cost +t sint),

11.3. y = 4(sint −t cost), 0 ≤t ≤ 2π.

x =(t2 −2)sint +2t cost,y =(2 −t2 )cost +2t sint,

0 ≤t ≤π.

11.5.y =10sin3 t, 0 ≤t ≤π2.

x =3(t −sint),

11.7.y =3(1−cost),

π≤t ≤ 2π.

x =3(cost +t sint),

11.9.y =3(sint −t cost),x =10cos3

0 ≤t ≤π3.

x =(t2 −2)sint +2t cost,y =(2 −t2 )cost +2t sint,

0 ≤t ≤π3.

x = 6cos3 t,

11.11. y = 6sin3 t, 0 ≤t ≤π3.

11.4.

x = et (cost +sint),

11.6. y = et (cost −sint), 0 ≤t ≤π.

x = 12 cost − 14 cos2t,

1 1

11.8. y = 2 sint − 4 sin 2t,

π2 ≤t ≤ 2π3.

11.10.

+),

11.12.y = et (cost −sint),

π2 ≤t ≤π.x = et (cost sint

x = 2,5(t −sint),

11.13.y = 2,5(1−cost),

π2 ≤t ≤π.

x =3,5(2cost −cos2t),

y =3,5(2sint −sin 2t),

0≤t ≤π2.

x = 6(cost +t sint),

11.15.y = 6(sint −t cost),

0 ≤t ≤π.

x =(t2 −2)sint +2t cost,y =(2 −t2 )cost +2t sint,

0 ≤t ≤π2.

x =8cos3 t,

11.17. y =8sin3 t,

0≤t ≤π6.

x = 4(t −sint),

11.19.y = 4(1−cost),

π2 ≤t ≤ 2π3.

x = 2(2cost −cos2t),

y = 2(2sint −sin 2t), 0 ≤t ≤π3.

11.14.

11.16.

x = et (cost +sint),

11.18. y = et (cost −sint), 0 ≤t ≤ 2π.

11.20.

x =8(cost +t sint),

11.21. y =8

(sint −t cost),

11.22.

0 ≤t ≤π 4.

x =(t2 −2)sint +2t cost,

y =(2 −t2 )cost +2t sint,

0 ≤t ≤ 2π.

x = e

(

cost +sint

)

x = 4cos t,

11.24.

y = et (cost −sint),

11.23. y = 4sin3 t,

π 6 ≤t ≤π 4.

0 ≤t ≤3π 2.

x = 2

(t −sint),

11.25. y = 2

(1−cost),

11.26.

0 ≤t ≤π 2.

x = 4

(2cost −cos2t),

y = 4

(2sint −sin 2t),

0 ≤t ≤π.

x = 2

(cost +t sint),

11.27. y = 2

(sint −t cost),

11.28.

0 ≤t ≤π 2.

x =(t2 −2)sint +2t cost,

y =(2 −t2 )cost +2t sint,

0 ≤t ≤3π.

x = e

cost +sint

(

)

x = 2cos

11.30.

y = et (cost −sint),

11.29. y = 2sin3 t,

0 ≤t ≤π 4.

π 6 ≤t ≤π 4.

x =(t2 −2)sint +2t cost,

11.31.y =(2 −t2 )cost +2t sint,

0 ≤t ≤π.

Задача 12. Вычислить длины дуг кривых, заданных уравнениями в полярных координатах.

12.1.ρ =3e3ϕ4 , −π2 ≤ϕ ≤π2.

12.2.ρ = 2e4ϕ3 , −π2 ≤ϕ ≤π2.

12.3.ρ = 2 eϕ , −π2 ≤ϕ ≤π2.

12.4.ρ =5e5ϕ12 , −π2 ≤ϕ ≤π2.

12.5.ρ = 6e12ϕ5 , −π2 ≤ϕ ≤π2.


12.6. ρ =3e3ϕ 4 ,				0 ≤ϕ ≤π 3.
12.7. ρ = 4e4ϕ 3 ,				0 ≤ϕ ≤π 3.
12.8. ρ =			eϕ ,	0 ≤ϕ ≤π 3.
	2
12.9. ρ =5e5ϕ 12 ,				0 ≤ϕ ≤π 3.
12.10. ρ =12e12ϕ 5 ,					0 ≤ϕ ≤π 3.
12.11. ρ =1−sinϕ,					−π 2 ≤ϕ ≤ −π 6.
12.12. ρ = 2		(1−cosϕ),				−π ≤ϕ ≤ −π 2.
12.13. ρ =3(1+sinϕ),						−π 6 ≤ϕ ≤ 0.
12.14. ρ = 4		(1−sinϕ),				0 ≤ϕ ≤π 6.
12.15. ρ =5		(1−cosϕ),				−π 3 ≤ϕ ≤ 0.
12.16. ρ = 6		(1+sinϕ),				−π 2 ≤ϕ ≤ 0.
12.17. ρ = 7		(1−sinϕ),				−π 6 ≤ϕ ≤π 6.
12.18. ρ =8		(1−cosϕ),				−2π 3 ≤ϕ ≤ 0.

12.19.ρ = 2ϕ, 0 ≤ϕ ≤34.

12.20.ρ = 2ϕ, 0 ≤ϕ ≤ 43.

12.21.ρ = 2ϕ, 0 ≤ϕ ≤512.

12.22.ρ = 2ϕ, 0 ≤ϕ ≤125.

12.23.ρ = 4ϕ, 0 ≤ϕ ≤34.

12.24.ρ =3ϕ, 0 ≤ϕ ≤ 43.

12.25.ρ =5ϕ, 0 ≤ϕ ≤125.

12.26. ρ = 2cosϕ,	0 ≤ϕ ≤π 6.
12.27. ρ =8cosϕ,	0 ≤ϕ ≤π 4.
12.28. ρ = 6cosϕ,	0 ≤ϕ ≤π 3.
12.29. ρ = 2sinϕ,	0 ≤ϕ ≤π 6.
12.30. ρ =8sinϕ,	0 ≤ϕ ≤π 4.
12.31. ρ = 6sinϕ,	0 ≤ϕ ≤π 3.

Задача 13. Вычислить объемы тел, образованных вращением фигур, ограниченных графиками функций. В вариантах 1–16 ось вращения Ox , в вариантах 17–31 ось вращения Oy .


13.1. y = −x2 +5x −6,				y = 0.
13.2. 2x − x2 − y = 0, 2x2 −4x + y = 0.
13.3. y =3sin x,			y =sin x,		0 ≤ x ≤π.
13.4. y =5cos x,			y = cos x,		x = 0, x ≥ 0.
13.5. y =sin2 x,		x =π 2,			y = 0.
13.6. x = 3			x =1,	y =1.
	y −2,
13.7. y = xex ,		y = 0,		x =1.
13.8. y = 2x − x2 ,			y = −x +2, x = 0.
13.9. y = 2x − x2 ,			y = −x +2.

13.10. y = e1−x , y = 0, x = 0, x =1.

13.11.y = x2 , y2 − x = 0.

13.12.x2 +(y −2)2 =1.

y =1− x2 ,

x = 0,

x =

x =1.

13.13.

y −2,

13.14.

y = x2 ,

y =1,

x = 2.

y = x3 ,

y =

13.15.

13.16.

y =sin(πx 2),

y = x2.

13.17.

y = arccos(x 3),

y = arccos x,

y = 0.

13.18.

y = arcsin(x 5),

y = arcsin x,

y =π 2.

13.19.

y = x2 ,

x = 2,

y = 0.

13.20.

y = x2 +1,

y = x,

x = 0,

y = 0.

y =

y = 0,

y =1,

x = 0,5.

13.21.

x −1,

13.22.

y = ln x,

x = 2,

y = 0.

13.23.

y =(x −1)2 ,

y =1.

13.24.

y2 = x −2,

y = 0,

y = x3 ,

y =1.

13.25.

y = x3 ,

y = x2.

13.26.

y = arccos(x 5),

y = arccos(x 3), y = 0.

13.27.

y = arcsin x,

y = arccos x,

y = 0.

13.28.

y = x2 −2x +1,

x = 2,

y = 0.

13.29.

y = x3 ,

y = x.

13.30.

y = arccos x,

y = arcsin x,

x = 0.

13.31.

y =(x −1)2 ,

x = 0,

x = 2,

y = 0.

<<< < Предыдущая 1 23 / 33

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