Multivariable Calculus (Math 214) Assignment

due November 8, 1999

  1. Let F be the vector field (vector-valued function of a point) in the plane that is defined away from the origin by

    F(x, y) = ({-y}/{SQRT{x^{2} + y^{2}}}, {x}/{SQRT{x^{2} + y^{2}}}) .

    Find the path integral of this vector field over the path C that is given parametrically by

    R(t) = (a cos t, a sin t) for 0 <= t <= theta .

  2. Do exercise 15 on p. 836: the double integral of the function

    f(x, y) = x + y

    taken over the triangular region in the (x, y) plane with vertices at (0, 0), (0, 1), and (1, 1).

  3. Find the double integral of the function f(x, y) = x^{2} y + x y^{3} over the planar region that is bounded by the parabola y = 4 - x^{2} and the line y = -3 x.

  4. Reverse the order of iteration in the double integral

    INT[_{0}^{2} INT[_{0}^{e^{x}} dy ] dx ] .

    What does this integral represent?

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