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Integrals

Derivatives and Integrals : Area and Volume of Solid

Explain why the derivative function of the function g(x) = x is equal to 1 on the interval (0,∞), equal to 1 on the interval (-∞,0), and undefined at 0. [Hint. Sketch the graph of g.] Consider the region R bounded by the curve xy =3 and the lines x I and x =4. Set up the integrals (do not evaluate) that give the

Stokes' Theorem : Curls and Surface Integrals

Let F = (2x, 2y, 2x + 2z). Use Stokes' theorem to evaluate the integral of F around the curve consisting of the straight lines joining the points (1,0,1), (0,1,0) and (0,0,1). In particular, compute the unit normal vector and the curl of F as well as the value of the integral:

Integration Word Problems : Rate of change

Suppose that a tank initially contains 2000 gal of water and the rate of change of its volume after the tank drains for t min is '(t)=(0.5)t)-30 (in gallons per minute). How much water does the tank contain after it has been draining for 25 minutes? keywords: integration, integrates, integrals, integrating, double, triple, m

Volume of a solid

The region R is bounded by the graphs of x-2y = 3 and x=y^2. Set up (but not evaluate) the integral that gives the volume of the solid obtained by rotating R around the line x=-1.

Integrals and Average Sums

Please see the attached file for the fully formatted problems. keywords: integration, integrates, integrals, integrating, double, triple, multiple

Solids

The region bound by the circle (x-a)^2 + y^2 = a^2 is revolved around about the y-axis to generate a solid 1 find the volume 2 find the surface area

Integration and Sums of Rectangles

1. Given f ′(x) = ex + (4/3)x^(-2/3) , find f (x) if f(1) = e. 2. Given f (x) = x2 + x +1 (a) Approximate the area between the curve of f and the x-axis on the interval [0,2] using 4 rectangles and right point sums. (b) Find the EXACT area between the curve of f and the x-axis on the interval [0,2] by using area a

Definite Integrals

1) 3x+6/(x^2=4x=5)^2 dx --This is a definite integration problem evaluated at b (1) and a (-1) 2) ln x/x dx --Evaluated at b (e) and a (1) 3) x^1/2 + 3x^1/3 + 3 dx --This is an indefinite integration problem. 4) e^3x + 3/x dx --Indefinite 5) e^-5x + e^5x dx --Definite-Evaluated at b (1) a (-1)

Plane Triangular Surface and Stokes' Theorem

4. Consider the plane triangular surface formed by the intersection of the plane x/A + y/B + z/C = 1 (A, B, and C all positive), with outward pointing normal, ie the normal pointing away from the origin. Verify Stokes' Theorem for the vector field F = (x + y) + (2x − z) + (y + z) by performing the surface integral a

Surface Integral of a Paraboloid of Revolution

Let S be the closed surface of the paraboloid of revolution z = ±(4 − x2 − y2 ) where −2 x, y +2. Evaluate the following surface integral directly and then by using the divergence theorem; where R is the position vector to a point on the surface and is the outward pointing normal at that point. See att

Integrating Differential Equations

1. Consider the following differential equation: (1-c/r)(dt/dl)2 - (1-c/r)-1(dr/dl)2 = 0 (a) Show that this equation can be written as dr/dt = (1-c/r) (b) Solve the above equation for t(r). Please evaluate integrals by hand. Take c to be a constant Please see the attached file for the fully formatted proble

Evaluate Integrals

The problems in the file submitted are from an undergraduate course in real Analysis. If you are able to work the problems, please detail any theorems or lemmas used in your solutions. The book we are using is titled "The Elements of Real Analysis" by Robert G. Bartle. We are working on derivatives and integrals, but have not

Integration and Limits Proof

If a>0 show that pi lim ∫ sin(nx)/nx dx = 0 a What happens if a = 0. The problem in the file submitted is from an undergraduate course in Real Analysis. If you are able to work the problems, please detail any theorems or lemmas used in your solutions.

Convergence or Divergence of Integrals

The problems in the file submitted are from an undergraduate course in real Analysis. If you are able to work the problems, please detail any theorems or lemmas used in your solutions. The book we are using is titled "The Elements of Real Analysis" by Robert G. Bartle. We are working on derivatives and integrals, but have not

Integrals : Mean Value

Please see the attached file for the fully formatted problems. keywords: integration, integrates, integrals, integrating, double, triple, multiple

Double Integrals and Integral of a Disk

Suppose that the double integral of f(x, y) dA = 4, where D is the disk x^2 + y^2 <= 1. Now suppose E is the disk x^2 + y^2 <= 9 and 9(x,y) = 3f(x/3, y/3). What is the value of the double integral of g(x, y) dA? Please see the attached file for the fully formatted problems. keywords: integration, integrates, integrals, in

Taylor Polynomials and Partial Sums of Series

1 Write the Taylor polynomial with center zero and degree 4 for the function f(x) = e^-x 2 Determine the values of p for which the series &#8734; &#931; 1/(2p)&#8319; n=1 3 Calculate the sum of the first ten terms of the series, then estimate the error

Trapezoid Reduction Formula

Approximate the integral using the trapezoid reduction formula with m=4. (Do by hand). Find the exact value of the integral and the exact error. See attached file for full problem description. keywords: integration, integrates, integrals, integrating, double, triple, multiple

Romberg Integration

2 Use Romberg integration and compute A33 to approximate the integral &#8747; x ln x dx where m=3. Please show all work.

Simpson's Rule and Composite Simpson's Rule and Errors

(4.3) 4d Find a bound for the error using Simpson's rule and compare this to the actual error for the following integrals (4.4) 2b Use the composite Simpson's rule to approximate the following integrals. , n = 4 Please see the attached file for the fully formatted problems.

Complex Integration : Holomorphisms

Let f(z) be holomorphic on the unit disc and f(0)=1. By working with 1/2ipi(integral over unit circle of [2+,-(z+1/z)]f(z) dz/z) prove that a)2/pi(integral(0 -2pi) of f(e^itheta)cos^2theta/2 d(theta))=2 + f'(0) b)2/pi(integral(0-2pi) of f(e^itheta)sin^2theta/2 d(theta))=2-f'(0)