### Limits : Summation Series

An = 1/ (n+1) + 1/(n+2) + 1/ (n +3) +......+ 1/(n+n) Prove the limit of the sequence exists (or not). (Question also included in attachment)

Explore BrainMass

- Anthropology
- Art, Music, and Creative Writing
- Biology
- Business
- Chemistry
- Computer Science
- Drama, Film, and Mass Communication
- Earth Sciences
- Economics
- Education
- Engineering
- English Language and Literature
- Gender Studies
- Health Sciences
- History
- International Development
- Languages
- Law
- Mathematics
- Philosophy
- Physics
- Political Science
- Psychology
- Religious Studies
- Social Work
- Sociology
- Statistics

An = 1/ (n+1) + 1/(n+2) + 1/ (n +3) +......+ 1/(n+n) Prove the limit of the sequence exists (or not). (Question also included in attachment)

Find the integral f(x) = kx(1-x)^2 where 'k' is a constant

Evaluate the given integral by changing to polar coordinates: Above the cone z = sqrt(x^2 +y^2) and below the sphere x^2 + y^2 +z^2 = 1 Please show steps, especially how you determine the boundaries. Thanks.

Please see the attached file for the fully formatted problems. Derive the source solution by performing integral transforms of the equation:

What is the integral of: Integral (2 --- 0) of x^2/ (4 +x^2)dx = ? A. 2 - ln(2) B. ln(2) - 1/2 C. 2 tan^2(2) + 4 ln[cos(2)] D. 2 ln[sec(2)]-sin^2(2) E. 2 - pi/2

What is the average value of y = (2x+1)1/2 over the interval [4,12]? A. √ 3 - 1/3 B. 2(√ 3 - 1/3) C. 4 D. 49/12 E. √ 17 Please explain solution in detail.

26. Let S be the closed region in the first quadrant of the xy-plane bounded by y = sin(pi x/2) and y = x for 0 ≤ x ≤ 1. What is the volume of the closed region in R3 obtained by revolving S about the x-axis? A. 2 - (pi /2) B. pi /6 C. pi /3 D. pi /2 E. (2pi )/3

25.∫(x-8)/(x2 - 4x) dx 6 ---> 8 A. -(47/576) B. 1/6 C. ln (8/9) D. ln 2 E. ln (32/9) A. -31 B. -19 C. 11 D. 30 E. 49 Please explain in detail. Thanks.

Please solve the attached integral substitution problem {also attached: multiple choice options} Thank you.

Please see the attached file for the fully formatted problems. 21. The region S is bounded by y = x2 - 2x + 3, y = 0, x = 0, and x = 9. Which of the following is the approximation to the area of S obtained by computing the sum of the areas of the 3 inscribed rectangles with bases [0,3], [3,6], and [6,9] (lower Riemann sum)?

20. Let S be the closed region in the first quadrant of the xy-plane bounded by y = 6x2, y = 0, x = 0, and x = 1. What is the volume of the solid obtained by revolving S about the line x = -1? A. 3x B. 7x C. 36x /5 D. 8x E. 56x /5

19. ∫0 to (pi/4) of x²cos x dx Please see attachment for full question.

18. Let F(x) = ∫0 to x^1/3 (√1+t^4) dt Then F'(0) = A. 0 B. 1/3 C. 2/3 D. 1 E. Does not exist. Please see attachment for full question.

17. If ∫0to1f(x)dx = -1 and ∫0to1g(x)dx= 1 then ∫1to0g(x)dx - ∫0to1 2f(x)dx = A. -3 B. -1 C. 0 D. 1 E. 3 Please see attachment for full question.

16. ∫0 to 3 x/(√x+1) dx = A. 3/8 B. 2/3 C. 3/2 D. 9/4 E. 8/3

15. ∫1 to ∞ 1/ (e^x +1)dx = A. ln (1 + e-1) B. - ln (1 + e-1) C. ln (1 + e) D. arctan (e1/2) E. does not exist Please see attachment

14. ∫dx/(x^3) dx -1 --> 2 A. (1/12)ln 8 B. 3/8 C. û(5/12) D. ln 8 E. does not exist

13. What is the average value of the function f(x) = x sin(x2) over the interval [2,4]? A. sin 4 + 2 sin 16 B. -sin 4 + 2 sin 16 C. (cos 4 - cos 16)/2 D. (cos 4 - cos 16)/4 E. (-cos 4 + cos 16)/4

12. What is the area of the closed region bounded by y = x2 - |x| and the x-axis, between x = -1 and x = 1? A. 1/12 B. 1/6 C. 1/3 D. 2/3 E. 5/6

Please see the attached file for the fully formatted problems. 11. ∫ x lnx dx 3 -->1 A. -2 + (9/4)ln 3 B. -4 + (9/2)ln 3 C. -(1/4) + (9/4)ln 3 D. -(5/2) + (9/2)ln 3 E. -2 + (9/2)ln 3 Please explain your answer in detail. Thanks.

Please see the attached file for the fully formatted problem. 10. ∫(x2 + 3x - 5)/x2 dx 6-->1 A. 5/6 - 3 ln 6 B. 61/6 + 3 ln 6 C. 5/6 + 3 ln 6 D. 17/6 + 3 ln 6 E. 17/6 - 3 ln 6

7. The closed region in the first quadrant bounded by the curves y = x3 and y = x(1/3) is rotated about the x-axis. What is the volume of the resulting solid? A. 1/2 B. 128x /455 C. 16x /35 D. x /2 E. 32x /35

What is the area of the closed region bounded by x = -1, x = 0, y = x2, and y = x3 ? A. 1/12 B. 1/6 C. 1/4 D. 5/12 E. 7/12

2. What is the average value of xe^x2 on the interval [2, 4]? A. (e16 - e4)/4 B. (e16 - e4)/2 C. (2e16 - e4)/2 D. 2e16 - e4 E. 2e16 + e4

Finding formulas for the volume enclosed by a hypersphere in n-dimensional space. c) Use a quadruple integral to find the hypervolume enclosed by the hypersphere x^2 + y^2 + z^2 + w^2 = r^2 in R^4. (Use only trigonometric substitution and the reduction formulas for ∫sin^n(x)*dx or ∫cos^n(x)*dx.)

Integrate the function (1/θ)^2 e^((-x1-x2)/ θ)) from 0 to 2 ln 2- x2 (for dx1) and from 0 to 2 ln 2 (for dx2).

∫(1 + tanθ)^5sec^2θ dθ

∫ √x sin(1 + x^3/2)dx (Corrected)

∫sec2θ tan2θ dθ

∫y^3 √(2y^4 -1)