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 15.1, pages 829–832:
   1. Let R = [0, 1] × [0, 2]. Find ∫ ∫ _Re^2x+3ydxdy.
   2. Let R = [1, 2] × [2, 3]. Find ∫ ∫ _Rx²y dxdy.
   3. Let R = [0, 1] × [0, 2]. Find ∫ ∫ _Rxe^xydxdy. (Hint: think about the order of integration.)
2. Section 15.2, pages 841–845:
   1. Integrate f(x,y) = (x + y + 1)^-2 over the triangle with vertices (0, 0), (0, 3), and (3, 0).
   2. Integrate f(x,y) = cos(2x + y) over the domain = {(x,y) : 0 ≤ x ≤, 0 ≤ y ≤ 2x}.
   3. Sketch the domain of integration of ∫ 01 ∫ exef(x,y)dy dx and express as an iterated integral in the opposite order.
   4. Let be the quadrilateral satisfying 1 ≤ y ≤ 2 and ≤ y ≤ x. Calculate the integral of f(x,y) = over . (Hint: think about the order of integration.)
3. Section 15.3, pages 854–856:
   1. Integrate f(x,y) = y over the domain = {(x,y,z) : x² + y² ≤ z ≤ 4}.
   2. Let be the region in the first quadrant {x ≥ 0,y ≥ 0,z ≥ 0} that lies below the paraboloid x² + y² = z - 2 and above the plane x + y + z = 2. Express ∫ ∫ ∫ f(x,y,z) dV as an iterated integral.
4. Section 15.4, pages 863–866: Express the region in Question 2d in polar coordinates. (You are not required to find the integral in polar coordinates.)

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).

• Suggestion 1 (free):

  a) write out the solutions neatly, and clearly, on regular paper

  b) use a pdf scanner on your phone, and produce one multipage pdf file. Recommended scanning apps include Adobe Scan (highly recommended), ABBYY FineScanner, CamScanner

• Suggestion 2 (not free):

  use an iPad, or something similar, along with a pdf producing note taking app (like Notability)

• Suggestion 3 (free):

  use LaTeX. This produces easy to read text but doing sketches is more complicated. The reason is that you will need to make pdf images of your sketches, and then include them in the LaTeX output. This is not difficult to do, and if you want to try this just ask and we will provide more information on what to do. Again, there is to be only one pdf file submitted per assignment.