Let C denote the line segment from z = i to z= 1. By observing that, of all the points on that line segment, the midpoint is the closest to the origin, show that |∫c dz/z^4| ≤ 4 sqrt(2) without evaluating the integral. Please see the attached file for the fully formatted problems.
Please see the attached file for the fully formatted problems.
F(z) = z - 1 and C is the arc from z = 0 to z = 2 consisting of (a) the semicircle z = 1 - e^(iθ) (pi ≤ θ ≤ 2pi) (b) the segment 0 ≤ x ≤ 2 of the real axis. Find the integral ∫c f(z) dz for the two cases.
Please see the attached file for the fully formatted problems.
Please see the attached file for full problem description.
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
4. Let f(x) = 2x + 1 for 0 =< x =< 1. If the interval [0,1] is partitioned into 4 subintervals of equal length, then what is the smallest Riemann sum for f(x) and this partition? A. 7/4 B. 15/8 C. 2 D. 7/2 E. 7
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