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    Area Between Two Lines and The Fundamental Theorem of Calculus

    Find the area between the curves 2y+x=0, y^2 =x+3. Use the fundamental theorem of calculus to evaluate d/dx ∫ t sin t dt. Please see the attached file for the fully formatted problems. keywords: integration, integrates, integrals, integrating, double, triple, multiple

    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

    Proving Irrationality Log Base

    Show that log(r), where this is log base 10, is irrational when r is a positive rational that is not an integral power of 10 I have already proven that e^r is irrational for all rational numbers r

    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:

    Stokes Theorem to Calculate the Surface Integral of the Curl

    See attached file for full problem description. Use Stokes' theorem to evaluate the surface integral of the curl: where the vector field F(x,y,z) = -12yzi + 12xzj + 18(x^2+y^2)zk and S is the part of the paraboloid z = x^2 + y^2 that lies inside the cylinder x^2 + y^2 =1, oriented upward.

    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 20. The average sum is used to approximate to the indicated accuracy. How large must n be chosen for this to be true? Could someone please tell me what am I doing wron

    Unique Factorization Domain with Quotient Field

    Let R be an integral domain with quotient field F and let p (X) be a monic polynomial in R[X] : Assume that p (X) = a (X) b (X) where a (X) and b (X) are monic polynomials in F [X] of smaller degree than p (X) : Prove that if a (X)is not in R[X] then R is not a UFD(unique factorization domain). Deduce that Z[2sqrt2] is not a

    Integral domains and ideals

    Let R be an integral domain and suppose that every prime ideal in R is principal. This exercise proves that every ideal of R is principal. (a) Assume that the set of ideals of R that are not principal is nonempty and prove that this set has a maximal element. [Use Zorn's Lemma.] (b) Let I be an ideal which is maximal with re

    Volume and surface area of a solid

    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

    Integral problem

    I need a full explanation with calculations for this problem: The indefinite integral of sq rt ( 1 + x^2)

    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

    Gradient Vector Fields and Line Integrals

    Consider the vector field F = (x^2 + y^2, 8xy). Compute the line integrals and , where c1(t) = (t, t^2) and c2(t) =(t, t) for 0<=t<=1. Can you decide from your answers whether or not F is a gradient vector field? Why or why not?

    Line integral and gradient

    Suppose that If f(0,0,0) = -5, find f(1, 1, 3). Hint: As a first step, define a path from (0,0,0) to (1, 1, 3) and compute a line integral. Please see the attached file for the fully formatted problems.

    Definite Integrals Evaluated

    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 &#8722; 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 &#8722; x2 &#8722; y2 ) where &#8722;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

    Radial Force Fields and Work

    Let F be the radial force field F= xi + yj. Find the work done by this force along the following two curves, both which go from (0, 0) to (5, 25). If C is the parabola... If C is the straight line segment... See attached file for full problem description.

    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 = &#61617;(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

    Calculate the Integral Question

    Calculate the Integral. 8. Let R be the region bounded by the graphs of x + y = 1, x + y = 2, 2x - 3y = 2, and 3x - 3y = 5. Use change of variables, x = (1/c)(3u + v), y = (1/c)(2u - v), to evaluate the double integral of (2x - 3y)dA. 10. Using multiple integration and a convenient coordinate system to find the volume of

    Evaluating Integrals for Real Analysis

    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. The boo