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Homework Help: Help with integration

  1. Feb 10, 2004 #1
    Evaluate the triple integral (grad dot F dV) over the region x^2+y^2+z^2 less than or equal to 25 if F+(x^2+y^2+z^2)(xi+yj+zk), by doing either a volume or a surface integral, whichever is easiest!


    will each of the integral go from -5 to 5? what is the easiest way?
     
  2. jcsd
  3. Feb 11, 2004 #2

    HallsofIvy

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    You have a problem:

    If you are attempting a problem like this, you should already have learned how to set up a general triple integral!

    No, the three integrals do not all go from -5 to 5. That would be a rectangular solid. Your figure here is a sphere. If you wanted to integrate over the volume of the sphere, in cartesian coordinates, you would have, say, x form -5 to 5, y from -√(25-x2) to +√(25-x2), z from -√(25-x2-y2) to +√(25-x2-y2).
    You could also use spherical coordinates in which you would have ρ from 0 to 5, θ from 0 to 2π, φ from 0 to π or cylindrical coordinates: r from 0 to 5, θ from 0 to 2π, z from -√(25- r2) to √(25- r2).

    In this case, with F= (x^2+y^2+z^2)(xi+yj+zk), &del;.F (That is, by the way "del dot F", not "grad dot F": "grad" is specifically del of a scalar function), also called "div F" or the "divergence of F" is complicated. I would be inclined to use the divergence theorem: [tex]\int\int_T\int(\del . v)dV= \int_S (v.n)dS[/tex] where T is a three dimensional solid, S is the surface of the solid, and n is the unit normal vector to the surface at each point.

    Since the surface of the figure is given by x2+ y2+z2= 25, The (not-unit) normal vector is given by the gradient of the left hand side: 2xi+ 2yj+ 2zk. If we choose to do the integration in the xy-plane, we divide by the k coefficient:
    n dσ= ((x/z)i+ (y/z)j+ k)dydx. and integrate on the circle x2+y2= 25.

    I would be inclined to use polar coordinates to do that since, then, z= √(25- r2) and F= 25(rcosθi+rsinθj+ &radic(25- r2)k)
    (In order to get the entire surface of the sphere you would need to do z>0 and z< 0 separately. Alternatively, because of the symmetry, just multiply by 2.)
     
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