Copyright\begin{matrix}c \\ ◯ \end{matrix}2004 ONG TCV Scoala Virtuala a Tanarului Matematician http://math.ournet.md 1

Integrala Riemann

1. Sa se calculeze:

1.1. ∫_{−1}^{2} x^{2} dx.  1.2. ∫_{−1}^{2} \sqrt[3]{x} dx.
1.3. ∫_{−1}^{1} (4x^{3} − 3x^{2} + 2x − 1) dx. 1.4. ∫_{1}^{3} (x^{2} + x − 2) dx.
1.5. ∫_{0}^{\frac{π}{2}} sin x dx.  1.6. ∫_{0}^{π} cos x dx.
1.7. ∫_{0}^{\frac{π}{4}} \frac{x^{2}}{1 + x^{2}} dx.  1.8. ∫_{0}^{1} \frac{dx}{\sqrt{x^{2} + 1}}.
1.9. ∫_{0}^{1} e^{x} dx.  1.10. ∫_{− \frac{π}{4}}^{\frac{π}{4}} \frac{dx}{cos^{2} x}.
1.11. ∫_{− \frac{\sqrt{3}}{2}}^{\frac{1}{2}} \frac{dx}{\sqrt{1 − x^{2}}} .  1.12. ∫_{\frac{1}{e}}^{e^{2}} \frac{dx}{x}.
1.13. ∫_{0}^{1} \frac{dx}{1 + x^{2}} .  1.14. ∫_{0}^{\frac{π}{2}} x sin x dx.
1.15. ∫_{1}^{e} x ln x dx.  1.16. ∫_{2}^{3} \frac{x + 1}{x^{2}(x − 1)} dx.
1.17. ∫_{1}^{2} \frac{e^{\frac{1}{x^{2}}}}{x^{3}} dx.  1.18. ∫_{0}^{1} xe^{x} dx.
1.19. ∫_{−π}^{π} sin^{2} x dx.  1.20. ∫_{−π}^{π} cos^{2} x dx.
1.21. ∫_{e}^{e^{3}} \frac{dx}{x ln x}.  1.22. ∫_{0}^{\frac{π}{4}} tg^{3} x dx.

Copyright\begin{matrix}c \\ ◯ \end{matrix}2004 ONG TCV Scoala Virtuala a Tanarului Matematician http://math.ournet.md 2

2. Sa se calculeze ariile plane limitate de curbele:

2.1. f(x) = 3x − x^{2}, g(x) = 0.
2.2. f(x) = 4x − x^{2}, g(x) = 0.
2.3. f(x) = x^{2} + 1, g(x) = 2.
2.4. f(x) = x^{2}, g(x) = 4.
2.5. f(x) = x^{2}, g(x) = x + 2.
2.6. f(x) = x^{2} − x, g(x) = 3x.
2.7. f(x) = 2x − x^{2}, g(x) = x.
2.8. f(x) = (x − 1)^{2} + 2, g(x) = 3x − 1.
2.9. f(x) = x^{2}, g(x) = 2x − x^{2}.
2.10. f(x) = x^{2}, g(x) = 3x + 4.
2.11. f(x) = x^{3}, g(x) = \sqrt{x}.
2.12. f(x) = \frac{5}{x}, g(x) = 6 − x.
2.13. f(x) = x^{2}, g(x) = \sqrt[3]{x}.
2.14. f(x) = x^{2}, g(x) = 2\sqrt{2x}.
2.15. f(x) = −\sqrt{x}, g(x) = \sqrt{x}, x ∈ [0, 4].
2.16. f(x) = e^{x}, g(x) = e^{−x}, x ∈ [0, 1].
2.17. f(x) = ln x, g(x) = ln^{2} x.
2.18. f(x) = \frac{1}{4}|4 − x^{2}|, g(x) = 7 − |x|.
2.19. f(x) = 0, g(x) = −x + 2, h(x) = \sqrt{x}.
2.20. f(x) = \frac{1}{x}, g(x) = x, x = 2.
2.21. f(x) = sin x, g(x) = cos x, x ∈ [0, \frac{π}{4} ].
2.22. f(x) = x − \frac{π}{2} , g(x) = cos x, x = 0.
2.23. f(x) = sin^{2} x, g(x) = x sin x, x ∈ [0, π].
2.24. f(x) = sin 2x, g(x) = sin x, x ∈ [\frac{π}{3} , π].

Copyright\begin{matrix}c \\ ◯ \end{matrix}2004 ONG TCV Scoala Virtuala a Tanarului Matematician http://math.ournet.md 3

2.25. f(x) = tg x, g(x) = \frac{2}{3} cos x, x = 0.
2.26. f(x) = arcsin x, g(x) = arccos x, h(x) = 0.
2.27. f(x) = 2^{x−2} + 1, g(x) = 2^{2−x} + 1, h(x) = \frac{3}{2}.
2.28. f(x) = 2 − |2 − x|, g(x) = \frac{6}{|x + 1|}.

2.29. f(x) = | lg x|, g(x) = 0, x = \frac{1}{10}, x = 10.
2.30. f(x) = ln |1 + x|, g(x) = −xe^{−x}, x = 1.

3. Sa se calculeze ariile plane limitate de curbele:

3.1. ρ^{2} = a^{2} cos 2φ.
3.2. x = a cos t, y = b sin t.
3.3. ρ = 4 sin^{2} φ.
3.4. x = a cos^{3} t, y = a sin^{3} t.
3.5. ρ = a(1 + cos φ).
3.6. x = \frac{c^{2}}{a} cos^{3} t, y = \frac{c^{2}}{b} sin^{3} t, c^{2} = a^{2} − b^{2}.
3.7. ρ = 2 + cos φ.
3.8. x = \frac{1 − t^{2}}{(1 + t^{2})^{2}} , y = \frac{2at}{(a + t^{2})^{2}} .
3.9. ρ = a sin 2φ.
3.10. x = t − t^{2}, y = t^{2} − t^{3}.
3.11. ρ = a cos φ, ρ = a(cos φ + sin φ).
3.12. x = t^{2} − 1, y = t^{3} − t^{2}.
3.13. ρ = 2 − cos φ, ρ = cos φ.
3.14. x = \frac{t − t^{3}}{1 + 3t^{2}} , y = \frac{4t^{2}}{1 + 3t^{2}} .
3.15. ρ = 2\sqrt{3} cos φ, ρ = 2 sin φ.
3.16. x = sin 2t, y = sin t.
3.17. ρ = 1 + \sqrt{2} cos φ.

3.18. x = 1 + t − t^{3}, y = 1 − 15t^{2}.
3.19. ρ = 3 sin φ, ρ = 5 sin φ.
3.20. x = 1 + 2 cos t, y = tg t + 2 sin t.

4. Sa se calculeze lungimile arcelor:

4.1. f(x) = \frac{(x + 1)^{2}}{4} − \frac{ln (x + 1)}{2}, x ∈ [0, 1].
4.2. f(x) = − ln cos x, x ∈ [0, \frac{π}{6} ].
4.3. f(x) = ln x, x ∈ [\sqrt{3}, \sqrt{8}].
4.4. f(x) = ln (x^{2} − 1), x ∈ [2, 3].
4.5. f(x) = \sqrt{2x − x^{2}} − 1, x ∈ [\frac{1}{4}, 1].
4.6. f(x) = x^{2}, x ∈ [0, 1].
4.7. f(x) = 4\sqrt{x − 1}, x ∈ [1, 2].
4.8. f(x) = x^{2} − ln \sqrt{x}, x ∈ [1, 2].
4.9. f(x) = x\sqrt{x}, x ∈ [0, 9].
4.10. f(x) = ln sin x, x ∈ [\frac{π}{3} , \frac{2π}{3} ].
4.11. x = a cos^{3} t, y = a sin^{3} t, t ∈ [0, 2π].
4.12. ρ = 2 sin φ.
4.13. x = 3(2 − t^{2}), y = 4t^{3}, x ⩾ 0.
4.14. ρ = cos^{3} \frac{φ}{3}.
4.15. x = cos^{4} t, y = sin^{4} t, t ∈ [0, \frac{π}{2} ].
4.16. ρ = a(1 − cos φ).
4.17. x = 6 cos^{3} t, y = 6 sin^{3} t, t ∈ [0, \frac{π}{3} ].
4.18. ρ = sin 3φ.
4.19. x = 2(t − sin t), y = 2(1 − cos t), t ∈ [0, \frac{π}{2} ].

4.20. ρ = \frac{1}{2} + sin φ.
4.21. x = e^{t}(cos t + sin t), y = e^{t}(cos t − sin t), t ∈ [\frac{π}{6} , \frac{π}{4} ].
4.22. ρ = cos φ − sin φ.
4.23. x = 2(cos t + t sin t), y = 2(sin t − t cos t), t ∈ [0, π].
4.24. ρ = 2 sin 4φ.

5. Sa se calculeze volumul corpului obtinut prin rotatia in jurul axei OX a suprafetei margi-

nite de curbele:

5.1. f(x) = −x^{2} + 7x − 12, g(x) = 0.
5.2. f(x) = \frac{4}{x}, g(x) = 0, x = 1, x = 4.
5.3. f(x) = 2x + \sqrt{2x}, g(x) = 0, x = 2 x = \frac{9}{2}.
5.4. f(x) = 2x − x^{2}, g(x) = 2 − x.
5.5. f(x) = arcsin x, x = 0, x = 1.
5.6. f(x) = xe^{x}, g(x) = 0, x = 1.
5.7. f(x) = x^{2}, g(x) = 0, x = 3.
5.8. f(x) = (x − 2)^{2}, g(x) = 4.
5.9. f(x) = e^{2−x}, g(x) = 0, x = 1, x = 2.
5.10. f(x) = e^{x}, g(x) = 0, x = 0, x = 1.
5.11. f(x) = 3 sin x, g(x) = sin x, x = 0, x = π.
5.12. f(x) = sin x, g(x) = 0, x = \frac{π}{6} , x = \frac{π}{2} .
5.13. f(x) = 4 − x^{2}, g(x) = 3x, x = −2, x = 0.
5.14. f(x) = \sqrt{x}e^{−x}, g(x) = 0, x = 1.
5.15. f(x) = sin^{2} x, g(x) = x sin x, x = 0, x = π.
5.16. f(x) = sin 2x, g(x) = 0, x = 0, x = \frac{π}{4} .
5.17. f(x) = 3x − x^{2}, g(x) = 0.
5.18. x = a cos^{3} t, y = a sin^{3} t.
5.19. f(x) = x^{2}, g(x) = \sqrt{x}.
5.20. f(x) = x^{3}, g(x) = x^{2}.