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Find the mass and center of gravity of the lamina lamina with density δ(x, y) = x + y is bounded by the x-axis, the
line x = 1, and the curve y =
√
x..
Perform the integration by transforming the ellipsoidal region of integration into a spherical region of integration and
then evaluating the transformed integral in spherical coordinates.
(a) RRR
G
x
2 dV , where G is the region enclosed by the ellipsoid 9x
2 + 4y
2 + z
2 = 36.
(b) RRR
G
(y
2 + z
2
) dV , where G is the region enclosed by the ellipsoid x
2
a
2
+
y
2
b
2
+
z
2
c
2
= 1
Use the transformation x = u/v, y = uv to evaluate the integral sum Z 2
1
Z y
1/y
(x
2 + y
2
) dxdy +
Z 4
2
Z 4/y
y/4
(x
2 + y
2
) dxdy
Find the Jacobian ∂(x, y, z)/∂(u, v, w).
(a) u = xy, v = y, w = x + z (b) u = x + y + z, v = x + y − z, w = x − y + z
For the following find a transformation u = f(x, y), v = g(x, y) that when applied to the region R in the xy-plane has
as its image the region S in the uv-plane.
Use cylindrical coordinates to find the volume of the following solids.
(a) The solid that is inside the surface r
2 + z
2 = 20 but not above the surface z = r
2
.
(b) The solid enclosed between the cone z = (hr)/a and the plane z = h.
Use a triple integral to find the volume of the wedge in the first octant that is cut from the solid cylinder y
2 + z
2 ≤ 1
by the planes y = x and x = 0
Evaluate the following improper integrals as iterated integrals.
(a) Z ∞
1
Z 1
e−x
1
x
3y
dy dx (b) Z 1
−1
Z √ 1
1−x2
√−1
1−x2
(2y + 1) dy dx
By changing the order of integration, show that : Z x
0
Z u
0
e
m(x−t)
f(t) dt du =
Z x
0
(x − t) e
m(x−t)
f(t) dt.
Show, by changing to polar coordinates, that Z a sin β
Z
√
a2−y2
y cot β
ln (x
2 + y
2
) dx dy = a
2β(ln a − 1/2), where a > 0 and
0 < β < π/2. Rewrite the Cartesian integral with the order of integration reversed.
