Solutions Final Examination Calculus 4
Find S F n S where FHx, y, zL = x2 z3 + 2 x y z3 + x z4 and S is the box with vertices H 1, 2, 3L.
1.
8- 1, 2, 3<
81, 2, 3<
8- 1, - 2, 3<
81, - 2, 3<
8- 1, 2, - 3<
81, 2, - 3<
8- 1, - 2, - 3<
81, - 2, - 3<
SOLUTION.
Solutions Final Examination Calculus 4
1. Find the flux of F = x z + y z + 3 z2 through the surface of the solid bounded by the paraboloid z = x2 + y2 and the
plane z = 1. HINT: Divergence Theorem.
z
y
x
SOLUTION. We apply the Divergence Theorem. The dive
Solutions Examination 3 Calculus 4
1a.
x
y
z
1
1
1
Write the limits of integration on the integral x 2 y 2 z 2 f Hx, y, zL z y x for the region in the first octant bounded by
the tetrahedron in the diagram.
SOLUTION. The z, y, x limits are computed in thi
Solutions Examination 2 Calculus IV
1.
Find the surface area of the part of the sphere x2 + y2 + z2 = 2 that lies inside the paraboloid z = x2 + y2 .
SOLUTION. The intersection of the two surfaces is given by
2
x2 + y2 + z2 = x2 + y2 + Ix2 + y2 M = 2
or
2
1a.
Find the gradient of f Hx, yL = sin Ix2 yM at the point PJp,
1
N
p
in the direction of the vector v = + 2 .
SOLUTION. The gradient is
1
f Jp, p N = I2 x y cos Ix2 yM, x2 cos Ix2 yMM
b.
= H2 cos p, p2 cos pN = -I2, p2 M.
Hp,1pL
Find the directional de