MATH 2010 Multivariable Calculus and Matrix Algebra, Homework 4.

Due: Monday, June 8, 2020, 11.59pm.

Answer the following questions. Upload your solutions as a single pdf file to LMS. Include your last name and HW1 in the name of the file, eg, mitchellHW1.pdf. No late homework accepted.

  1. Section 15.4, pages 863–866:
    1. Let the domain D = {(x,y) : x 0, x2 + y2 4}. Find Dxy dxdy using polar coordinates.
    2. Find 02 x2xy dy dx using polar coordinates. (Hint: the integral of 1 cos 2(θ) is tan(θ).)
    3. Let the domain D = {(r,θ) : 0 θ π∕2, 0 r ∘-------
 sin(2θ)}. Find Dx√ -2----2-
  x  + y dxdy using polar coordinates. (Hint: use double angle formula and also the identity sin 2θ + cos 2θ = 1.
    4. Use cyclindrical coordinates to integrate f(x,y,z) = z√ -2----2-
  x  + y over the domain W = {(x,y,z) : x2 + y2 z 2 - x2 - y2}.
    5. Use spherical coordinates to integrate f(x,y,z) = z over the domain W = {(x,y,z) : x2 + y2 + z2 4z, z √--------
 x2 + y2}.
  2. Section 15.5, pages 874–878:
    1. Find the total population within a 4-km radius of the origin with x 0, y 0, with population density 2000(x2 + y2)-0.3. (Hint: use polar coordinates.)
    2. Find the centroid of the infinite lamina with x 1 and 0 y x-3 with density δ(x,y) = 1.
    3. Find the centroid of W = {(x,y,z) : x,y,z 0, x∕R + z∕H 1, x2 + y2 R2} for positive parameters R and H. (Hint: use cylindrical coordinates and double angle formulas.)
    4. Let R be infinite lamina with x 0 and y 0 and with density δ(x,y) = e-x-y. Find the moments of inertia Ix and Iy.
    5. The random variables x and y have domain D = {(x,y) : 0 x 1, 0 y 1 -x}and probability density p(x,y) = Cxy on D for some positive constant C.
      1. Determine C so that p(x,y) is a probability density on D.
      2. What is the probability that x 2y?

Homework submission guidelines

Submissions: you can make multiple submissions of a homework, but only the final submission will be graded. The single pdf file submitted will be a multipage document, and the pages must be in the correct order.

Producing a pdf file: the homework assignments will involve extensive mathematical formulas and expressions, and will often require sketches (of curves, domains, etc).

    John Mitchell
    Amos Eaton 325
    mitchj at rpi dot edu
    Office hours: Mon, Tues, Thurs, Fri: 2.30–3.30pm, WebEx:
    TA: Rachel Wesley.
    Office hours: available on Slack and WebEx.
    WebEx time: Mon, Thurs 11am-noon, or by appointment.