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Integrals

Multiple Intergration, Area, Center of Mass, Moment, Centroid and Jacobian

1.Given the region R bounded by y=2x+2 , 2y=x and 4. a) Set up a double integral for finding the area of R. b) Set up a double integral to find the volume of the solid above R but below the surface f(x,y) 2+4x. c) Setup a triple integral to find the volume of the solid above R but below the surface f(x,y)=-x^2 +4x. d) Set

Evaluate the path integral of a helix.

Let C be the helix, with parameterization r(t)=(cost, sint, t), tE[0,2pi] and let f(x,y,z) = x^2 + y^2 + z^2. Evaluate the path integral. (See attachment for full question)

Integral of a Principal Branch

Show that the integration from -1 to 1 z^i dz = ((1+e^-pi)/2)*(1-i)where zi denotes the principal branch... (See attachment for full question)

Double Integral : Disc

By changing variables to polar coordinates evaluate the integral , where And , i.e., the disc of radius 3 centred at the origin. Please see the attached file for the fully formatted problems.

Double Integral : Change of Variables to Polar Coordinate

Prove ∫ 0 --->∞ e^(-x^2) dx = sqrt(pi)/2 Hint: multiply the integral with itself, use a different dummy variable y, say, for the second integral, write it as a double integral, and use change of variables to polar coordinate.

Euclid's Division Lemma and Fundamental Theorem of Arithmetic

1. Without assuming Theorem 2-1, prove that for each pair of integers j and k (k > 0), there exists some integer q for which j ? qk is positive. 2. The principle of mathematical induction is equivalent to the following statement, called the least-integer principle: Every non-empty set of positive integers has a least element.

Double Integral and Change of Order of Integration

I) Evaluate the integral.... ii) Change the order of integration and verify the answer is the same by evaluating the resulting integral. Please see the attached file for the fully formatted problems.

Analysis of a Midpoint of a Line Segment

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.

Integral of a Semicircle and Segment

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.

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)

Polar Coordinates

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.

Evaluating the Integral

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

Integrals : Volume of Solid of Revolution

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

Integrals

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.

Integral Substitution

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

Integrals : Lower Riemann Sum

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)?

Integrals : Volume of a Solid of Revolution

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

Integrals

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

Integrals

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.

Integrals

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.

Integrals

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

Integrals

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

Integrals

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