Math 32B Week 6 WorksheetWritten by Victoria Kala

February 12, 2019

1. Evaluate the integralRR

R e(x+y)/(x�y)dA where R is the trapezoidal region with vertices(1, 0), (2, 0), (0,�2), (0,�1). Use the transformation x = 1

2 (u + v), y = 12 (u � v). Hint: The

region of integration S in the uv-plane is {(u, v) : 1 v 2,�v u v}.

2. Let F = (yex + sin y)i+ (ex + x cos y)j.

(a) Calculate div(F).

(b) Determine whether or not F is a conservative vector field. If it is, find a potential functionf such that F = rf .

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3. Write the parametrization for the following problems. Hint : use polar coordinates for circleand disk parametrizations.

(a) The line from (1, 1) to (4, 2)

(b) The line from (4, 2) to (1, 1)

(c) The line from (2,�4, 5) to (7, 9,�1)

(d) The arc of the parabola x = 4� y2 from (�5,�3) to (0, 2)

(e) The upper half of the unit circle x2 + y2 = 1 starting at (1, 0) and ending at (�1, 0)

(f) The upper half of the unit circle x2 + y2 = 1 starting at (�1, 0) and ending at (1, 0)

(g) The unit circle centered at the origin, oriented clockwise

(h) The circle of radius 3 centered at (�1, 2)

(i) The disk x2 + y2 = 4 on the plane z = �5, oriented counterclockwise

(j) The disk x2 + z2 = 9 on the plane y = �1

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