Math 2300 (Fall 2014) Worksheet II Find the volume of the solid that lies between
Question # 00363863
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Updated on: 08/17/2016 01:10 AM Due on: 08/17/2016
Math 2300 (Fall 2014) Worksheet II
1. (triples integrals)
(a.) Find the volume of the solid that lies between the paraboloid z = x2 +y2 and the sphere x2 +y2 +z2 =
2. (Use cylindrical coordinates to evaluate the integral.)
(b.) Find the triple integral
(1 ? x2 ? y2 )dV
B
using spherical coordinates. Here B is the solid hemisphere x2 + y2 + z2 ? 1, and z ? 0.
2. (change of coordinates) Find the integral
(x ? y)dA,
R
using the coordinate change x = 2u + v, y = u + 2v. Here R is the triangular region with vertices (0, 0),
(2, 1), and (1, 2).
3. (Line integrals) Let F = (2x ? 3y)i + (?3x + 4y)j.
(a.) Determine if F is a conservative vector ?eld. If it is, ?nd its potential function.
(b.) Calculate the line integral
F · dr,
C
where C is the parametric curve given by
r(t) = 2 cos t, 2 sin t , 0 ? t ? ?.
4. (Green’s theorem) Use Green’s theorem to evaluate
C
F · dr, with
F := y cos x ? xy sin x, xy + x cos x ,
C is the triangle with vertices (0, 0), (0, 4), (2, 0).
5. (Curl and Divergence) Find the curl and the divergence of
F = x/y, y/z, z/x .
1
1. (triples integrals)
(a.) Find the volume of the solid that lies between the paraboloid z = x2 +y2 and the sphere x2 +y2 +z2 =
2. (Use cylindrical coordinates to evaluate the integral.)
(b.) Find the triple integral
(1 ? x2 ? y2 )dV
B
using spherical coordinates. Here B is the solid hemisphere x2 + y2 + z2 ? 1, and z ? 0.
2. (change of coordinates) Find the integral
(x ? y)dA,
R
using the coordinate change x = 2u + v, y = u + 2v. Here R is the triangular region with vertices (0, 0),
(2, 1), and (1, 2).
3. (Line integrals) Let F = (2x ? 3y)i + (?3x + 4y)j.
(a.) Determine if F is a conservative vector ?eld. If it is, ?nd its potential function.
(b.) Calculate the line integral
F · dr,
C
where C is the parametric curve given by
r(t) = 2 cos t, 2 sin t , 0 ? t ? ?.
4. (Green’s theorem) Use Green’s theorem to evaluate
C
F · dr, with
F := y cos x ? xy sin x, xy + x cos x ,
C is the triangle with vertices (0, 0), (0, 4), (2, 0).
5. (Curl and Divergence) Find the curl and the divergence of
F = x/y, y/z, z/x .
1
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Solution: Math 2300 (Fall 2014) Worksheet II Find the volume of the solid that lies between