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Unformatted text preview: MAT 397 REVIEW 3 11—26-07 . Evaluate this integral / / ——:—B—§ dA where D is the rectangle [0,1] x [0,4].
D (1 + my) . Evaluate this integral / / sin (y2) dA Where D is the triangle with vertices
(0,0), (0,2) and (4,2). 1 1
. Consider the integral / / f (ac, y) d3: dy.
0 y (a) Carefully sketch the region over which the integration is taking place.
(b) Write an equivalent double integral with the order of integration reversed. (c) Write an equivalent integral in polar coordinates. . Find the mass of the lamina which occupies the region inside the circle x2 + y2 : 2y
and outside the circle 11:2 + y2 = 1 and which has density equal to the reciprocal of its
distance from the origin. . Find the surface area of that part of the the paraboloid z = 4 - m2 — y2 that lies above
the z = 0 plane. 1 1 l—z
. Consider the inte ral/ / / f cc,y,z dz dzL' dy.
g 0 _2y_y2 O ( ) (a) Carefully sketch the solid over which the integration is taking place.
(b) Write the triple integral with the the order dac dz dy. (b) Write the triple integral with the the order dy dz dm. . Set up but do not evaluate the triple integral giving the volume of the solid in the
ﬁrst octant bounded by the surfaces 22 = x2 + y2 and z = x2 + y2 (a) In rectangular coordinates.
(b) In cylindrical coordinates.
(c) In spherical coordinates. ) (d Find the volume by evaluating one of these integrals, whichever you ﬁnd the
easiest. 8. Evaluate the integrals 3 JET—$7 9—:112—y2
(a) / / / 11:2 dz dy dIL'.
—3 —\/9—z2 0 3 «9‘37 m
(b) / / / 2V1? + 1/2 + 22 dz dy d1).
—3 -¢9—x5 0 9. By using the transformation u = w + y and 'u = x — y evaluate the integral
// (a: + 3/)6932‘3’2 dA where D is the region bounded by the lines at — y = 0, :13 — y =
1,m+y=0 andsc+y=4. ...
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- Fall '07