MA242 Review III

Double Integral

  • Definition:  Given a function f(x,y) and a domain.
  • Partition of the domain,  the norm of partition is ...
  • Approximate function by a constant
  • Form the Riemann Sum
  • Take the limit
    • If the limit exists, the integral is the limit of the Riemann sum. Otherwise, f is not integrable
     

    Evaluation  double integral.

  • Approximate method: Use the Riemann sum.

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  • Analytic:  Change to iterated integral. Important: The line test.
  • Fubini's theorem if the domain is a rectangle,  a <= x <= b,   c <= y <= d.

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  • Integrate with y first,  solve for y, if   g1(x) <= y <= g2(x).

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  • Integrate with x first,  solve for x, if   h1(y) <= x <= h2(y).

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  • Break-up the domain if the line test fails (a line that parallel to y axis hits the domain no more than twice if we integrate with y first).

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  • Use polar coordinates if the integrant or the boundaries of the domain contain  x2 + y2, circular domains, annulus or part of them. Pay attention  to some special curves in the polar coordinates, for example, the circle which is not centered at the origin.
  • Applications:
  • Area, f(x,y) = 1.

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  • Volume. Procedure.  Note volumes can also be calculated using triple integrals.
  • Solve for one z = f(x,y), the other surfaces do not contain z.  We can find the volume of the cylinder between  z=f(x,y)  and xy plane. Usually set z=0  or  z = C to find the projections of all surfaces on the xy-plane. Sketch the domain.
  • Determine whether we want to use xy- coordinates or polar coordinates.
  • If there are more than two surfaces that can be solved as z=f(x,y).  we need to break up as two separated ones and find the difference as net volume.
  • Total mass,  moments about x and y axes, center,  inertial about x and y axes and the origin of a laminar.

    • Triple  integrals.

  •     Analytic:  Change to iterated integral. Important: The line test.
  • Fubini's theorem if the domain is a rectangular box,  a <= x <= b,   c <= y <= d,  r <= z <= s.

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  •  Get rid of one variable to change it to a double integral.      g1(x,y) <= z <= g2(x,y),  or
    •             h1(x,z) <= y <= h2(x,z),
                q1(y,z) <= x <= q2(x,y),
  • Find the maximum projection on x-y plane if z is going to be integrated first. The projections are intersections of any of two surfaces by eliminating the z-variable if z is going to be integrated first.
  •  Volume of    a <= x <= b,   h1(x)<= y <= h2(x),  q1(x,y) <= z <= q2(x,y).

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  • Use cylindrical coordinates if the integrant or the boundaries of the domain contain  x2 + y2,  x2+z2,  y2+z2, circular domains, annulus, or cones, half planes, or part of them.
  •     Applications:
  •  Volume, f(x,y,z) = 1. Procedure.
  •  Total mass,  moments about x and y planes, center,  inertial about x and y plane and the  origin of a solid body.

    • Spherical Coordinates:

            x =
            y =                                                             Meanings of parameters.
            z =
  •     Special surfaces  or part of them in the spherical coordinates:
  •         Sphere                                                          First Octant
  •         cylinder                                                        Upper plane
  •         cones                                                             Half plane
  •     Jacobin

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  •     Differential area

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  •     General coordinates transform and its Jacobin.