# User:Melikamp/ma225

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## Parametric Equations

1. Find parametric equations describing the line segment from P(0,0) to Q(7,17).

x=7t and y=17t, where 0 ≤ t ≤ 1

2. Find parametric equations describing the line segment from ${\displaystyle P(x_{1},y_{1})}$ to ${\displaystyle Q(x_{2},y_{2})}$.

${\displaystyle x=x_{1}+(x_{2}-x_{1})t{\mbox{ and }}y=y_{1}+(y_{2}-y_{1})t,{\mbox{ where }}0\leq t\leq 1}$

3. Find parametric equations describing the ellipse centered at the origin with major axis of length 6 along the x-axis and the minor axis of length 3 along the y-axis, generated clockwise.

${\displaystyle x=3\cos(-t),\ y=1.5\sin(-t)}$

## Polar Coordinates

20. Convert the equation into Cartesian coordinates: ${\displaystyle r=\sin(\theta )\sec ^{2}(\theta ).}$

${\displaystyle y=x^{2}}$

21. Find an equation of the line y=mx+b in polar coordinates.

${\displaystyle r={\frac {b}{\sin(\theta )-m\cos(\theta )}}}$

Sketch the following polar curves without using a computer.

22. ${\displaystyle r=2-2\sin(\theta )}$
23. ${\displaystyle r^{2}=4\cos(\theta )}$
24. ${\displaystyle r=2\sin(5\theta )}$

Sketch the following sets of points.

25. ${\displaystyle \{(r,\theta ):\theta =2\pi /3\}}$
26. ${\displaystyle \{(r,\theta ):|\theta |\leq \pi /3{\mbox{ and }}|r|<3\}}$

## Calculus in Polar Coordinates

Find points where the following curves have vertical or horizontal tangents.

40. ${\displaystyle r=4\cos(\theta )}$

Horizontal tangents at (2,2) and (2,-2); vertical tangents at (0,0) and (4,0)

41. ${\displaystyle r=2+2\sin(\theta )}$

Horizontal tangents at (r,θ) = (4,π/2), (1,7π/6) and (1,-π/6); vertical tangents at (r,θ) = (3,π/6), (3,5π/6), and (0,3π/4)

Sketch the region and find its area.

42. The region inside the limaçon ${\displaystyle 2+\cos(\theta )}$
9π/2
43. The region inside the petals of the rose ${\displaystyle 4\cos(2\theta )}$ and outside the circle ${\displaystyle r=2}$
${\displaystyle 8\pi /3+4{\sqrt {3}}}$

## Vectors and Dot Product

60. Find an equation of the sphere with center (1,2,0) passing through the point (3,4,5)

${\displaystyle (x-1)^{2}+(y-2)^{2}+z^{2}=33}$

61. Sketch the plane passing through the points (2,0,0), (0,3,0), and (0,0,4)
62. Find the value of ${\displaystyle |\mathbf {u} +3\mathbf {v} |}$ if ${\displaystyle \mathbf {u} =\langle 1,3,0\rangle }$ and ${\displaystyle \mathbf {v} =\langle 3,0,2\rangle }$

${\displaystyle {\sqrt {145}}}$

63. Find all unit vectors parallel to ${\displaystyle \langle 1,2,3\rangle }$

${\displaystyle \pm {\frac {1}{\sqrt {14}}}\langle 1,2,3\rangle }$

64. Prove one of the distributive properties for vectors in ${\displaystyle \mathbb {R} ^{3}}$: ${\displaystyle c(\mathbf {u} +\mathbf {v} )=c\mathbf {u} +c\mathbf {v} }$

$eqnarray}"): {\displaystyle \begin{eqnarray} c(\mathbf u + \mathbf v) &=& c(\langle u_1, u_2, u_3\rangle + \langle v_1, v_2, v_3\rangle)\\ &=& c\langle u_1+v_1, u_2+v_2, u_3+v_3\rangle\\ &=& \langle c(u_1+v_1), c(u_2+v_2), c(u_3+v_3)\rangle\\ &=& \langle cu_1+cv_1, cu_2+cv_2, cu_3+cv_3\rangle\\ &=& \langle cu_1, cu_2, cu_3\rangle + \langle cv_1, cv_2, cv_3\rangle\\ &=& c\mathbf u + c\mathbf v. \end{eqnarray}$

65. Find all unit vectors orthogonal to ${\displaystyle 3\mathbf {i} +4\mathbf {j} }$ in ${\displaystyle \mathbb {R} ^{2}}$

${\displaystyle \pm \left\langle {\frac {-4}{5}},{\frac {3}{5}}\right\rangle }$

66. Find all unit vectors orthogonal to ${\displaystyle 3\mathbf {i} +4\mathbf {j} }$ in ${\displaystyle \mathbb {R} ^{3}}$

${\displaystyle \left\langle {\frac {4}{5}}c,-{\frac {3}{5}}c,{\sqrt {1-c^{2}}}\right\rangle ,\ c\in [-1,1]}$

67. Find all unit vectors that make an angle of ${\displaystyle \pi /3}$ with the vector ${\displaystyle \langle 1,2\rangle }$

${\displaystyle {\frac {\sqrt {5}}{10}}\left\langle 1\pm 2{\sqrt {3}},\ 2\mp {\sqrt {3}}\right\rangle }$

## Cross Product

Find ${\displaystyle \mathbf {u} \times \mathbf {v} }$ and ${\displaystyle \mathbf {v} \times \mathbf {u} }$

80. ${\displaystyle \mathbf {u} =\langle -4,1,1\rangle }$ and ${\displaystyle \mathbf {v} =\langle 0,1,-1\rangle }$

${\displaystyle \mathbf {u} \times \mathbf {v} =\langle -2,-4,-4\rangle }$

81. ${\displaystyle \mathbf {u} =\langle 1,2,-1\rangle }$ and ${\displaystyle \mathbf {v} =\langle 3,-4,6\rangle }$

${\displaystyle \mathbf {u} =\langle 8,-9,-10\rangle }$

Find the area of the parallelogram with sides ${\displaystyle \mathbf {u} }$ and ${\displaystyle \mathbf {v} }$.

82. ${\displaystyle \mathbf {u} =\langle -3,0,2\rangle }$ and ${\displaystyle \mathbf {v} =\langle 1,1,1\rangle }$

${\displaystyle {\sqrt {38}}}$

83. ${\displaystyle \mathbf {u} =\langle 8,2,-3\rangle }$ and ${\displaystyle \mathbf {v} =\langle 2,4,-4\rangle }$

${\displaystyle 6{\sqrt {41}}}$

84. Find all vectors that satisfy the equation ${\displaystyle \langle 1,1,1\rangle \times \mathbf {u} =\langle 0,1,1\rangle }$

None

85. Find the volume of the parallelepiped with edges given by position vectors ${\displaystyle \langle 5,0,0\rangle }$, ${\displaystyle \langle 1,4,0\rangle }$, and ${\displaystyle \langle 2,2,7\rangle }$

${\displaystyle 140}$

86. A wrench has a pivot at the origin and extends along the x-axis. Find the magnitude and the direction of the torque at the pivot when the force ${\displaystyle \mathbf {F} =\langle 1,2,3\rangle }$ is applied to the wrench n units away from the origin.

${\displaystyle \mathbf {\tau } =\langle 0,-3n,2n\rangle }$, so the torque is directed along ${\displaystyle \pm \langle 0,-3,2\rangle }$

Prove the following identities or show them false by giving a counterexample.

87. ${\displaystyle \mathbf {u} \times (\mathbf {u} \times \mathbf {v} )=\mathbf {0} }$

False: ${\displaystyle \mathbf {i} \times (\mathbf {i} \times \mathbf {j} )=-\mathbf {j} }$

88. ${\displaystyle \mathbf {u} \cdot (\mathbf {v} \times \mathbf {w} )=\mathbf {w} \cdot (\mathbf {u} \times \mathbf {v} )}$

Once expressed in component form, both sides evaluate to ${\displaystyle u_{1}v_{2}w_{3}-u_{1}v_{3}w_{2}+u_{2}v_{3}w_{1}-u_{2}v_{1}w_{3}+u_{3}v_{1}w_{2}-u_{3}v_{2}w_{1}}$

89. ${\displaystyle (\mathbf {u} -\mathbf {v} )\times (\mathbf {u} +\mathbf {v} )=2(\mathbf {u} \times \mathbf {v} )}$

$eqnarray}"): {\displaystyle \begin{eqnarray}(\mathbf u-\mathbf v)\times(\mathbf u+\mathbf v)&=&(\mathbf u-\mathbf v)\times\mathbf u + (\mathbf u-\mathbf v)\times\mathbf v\\&=&\mathbf u\times\mathbf u - \mathbf v\times\mathbf u + \mathbf u\times\mathbf v - \mathbf v\times\mathbf v\\&=&\mathbf u\times\mathbf v-\mathbf v\times\mathbf u\\&=&\mathbf u\times\mathbf v + \mathbf u \times \mathbf v\\&=&2(\mathbf u\times\mathbf v)\end{eqnarray}$

## Calculus of Vector-Valued Functions

100. Differentiate ${\displaystyle \mathbf {r} (t)=\langle te^{-t},t\ln t,t\cos(t)\rangle }$.

${\displaystyle \langle e^{-t}-te^{-t},\ln(t)+1,cos(t)-t\sin(t)\rangle }$

101. Find a tangent vector for the curve ${\displaystyle \mathbf {r} (t)=\langle 2t^{4},6t^{3/2},10/t\rangle }$ at the point ${\displaystyle t=1}$.

${\displaystyle \langle 8,9,-10\rangle }$

102. Find the unit tangent vector for the curve ${\displaystyle \mathbf {r} (t)=\langle t,2,2/t\rangle ,\ t\neq 0}$.

${\displaystyle \displaystyle {\frac {\langle t^{2},0,-2\rangle }{\sqrt {t^{4}+4}}}}$

103. Find the unit tangent vector for the curve ${\displaystyle \mathbf {r} (t)=\langle \sin(t),\cos(t),e^{-t}\rangle ,\ t\in [0,\pi ]}$ at the point ${\displaystyle t=0}$.

${\displaystyle \displaystyle {\frac {\langle 1,0,-1\rangle }{\sqrt {2}}}}$

104. Find ${\displaystyle \mathbf {r} }$ if ${\displaystyle \mathbf {r} '(t)=\langle {\sqrt {t}},\cos(\pi t),4/t\rangle }$ and ${\displaystyle \mathbf {r} (1)=\langle 2,3,4\rangle }$.

${\displaystyle \displaystyle \left\langle {\frac {2t^{3/2}+4}{3}},{\frac {\sin(\pi t)}{\pi }}+3,4\ln |t|+4\right\rangle }$

105. Evaluate ${\displaystyle \displaystyle \int _{0}^{\ln 2}(e^{-t}\mathbf {i} +2e^{2t}\mathbf {j} -4e^{t}\mathbf {k} )dt}$

${\displaystyle \langle 1/2,3,-4\rangle }$

## Motion in Space

120. Find velocity, speed, and acceleration of an object if the position is given by ${\displaystyle \mathbf {r} (t)=\langle 3\sin(t),5\cos(t),4\sin(t)\rangle }$.

${\displaystyle \mathbf {v} =\langle 3\cos(t),-5\sin(t),4\cos(t)\rangle }$, ${\displaystyle |\mathbf {v} |=5}$, ${\displaystyle \mathbf {a} =\langle -3\sin(t),-5\cos(t),-4\sin(t)\rangle }$

121. Find the velocity and the position vectors for ${\displaystyle t\geq 0}$ if the acceleration is given by ${\displaystyle \mathbf {a} (t)=\langle e^{-t},1\rangle ,\ \mathbf {v} (0)=\langle 1,0\rangle ,\ \mathbf {r} (0)=\langle 0,0\rangle }$.

${\displaystyle \mathbf {v} (t)=\langle 2-e^{-t},t\rangle }$, ${\displaystyle \mathbf {r} (t)=\langle e^{-t}+2t-1,t^{2}/2\rangle }$

## Length of Curves

Find the length of the following curves.

140. ${\displaystyle \mathbf {r} (t)=\langle 4\cos(3t),4\sin(3t)\rangle ,\ t\in [0,2\pi /3].}$

${\displaystyle 8\pi }$

141. ${\displaystyle \mathbf {r} (t)=\langle 2+3t,1-4t,3t-4\rangle ,\ t\in [1,6].}$

${\displaystyle 5{\sqrt {34}}}$

## Parametrization and Normal Vectors

142. Find a description of the curve that uses arc length as a parameter: ${\displaystyle \mathbf {r} (t)=\langle t^{2},2t^{2},4t^{2}\rangle \ t\in [1,4].}$

${\displaystyle \displaystyle \mathbf {r} (s)=\left({\frac {s}{\sqrt {21}}}+1\right)\langle 1,2,4\rangle }$

143. Find the unit tangent vector T and the principal unit normal vector N for the curve ${\displaystyle \mathbf {r} (t)=\langle t^{2},t\rangle .}$ Check that TN=0.

${\displaystyle \mathbf {T} (t)=\displaystyle {\frac {\langle 2t,1\rangle }{\sqrt {4t^{2}+1}}},\ \mathbf {N} (t)=\displaystyle {\frac {\langle 1,-2t\rangle }{\sqrt {4t^{2}+1}}}}$

## Equations of Lines And Planes

160. Find an equation of a plane passing through points ${\displaystyle (1,1,2),\ (1,2,2),\ (-1,0,1).}$

${\displaystyle x-2z+3=0}$

161. Find an equation of a plane parallel to the plane 2xy+z=1 passing through the point (0,2,-2)

${\displaystyle 2x-y+z+4=0}$

162. Find an equation of the line perpendicular to the plane x+y+2z=4 passing through the point (5,5,5).

${\displaystyle \mathbf {r} (t)=\langle 5+t,5+t,5+2t\rangle }$

163. Find an equation of the line where planes x+2yz=1 and x+y+z=1 intersect.

${\displaystyle \mathbf {r} (t)=\langle 1-3t,2t,t\rangle }$

164. Find the angle between the planes x+2yz=1 and x+y+z=1.

${\displaystyle \cos ^{-1}{\frac {2}{\sqrt {18}}}}$

165. Find the distance from the point (3,4,5) to the plane x+y+z=1.

${\displaystyle {\frac {11}{3}}{\sqrt {3}}}$

## Limits And Continuity

Evaluate the following limits.

180. ${\displaystyle \displaystyle \lim _{(x,y)\rightarrow (1,-2)}{\frac {y^{2}+2xy}{y+2x}}}$

−2

181. ${\displaystyle \displaystyle \lim _{(x,y)\rightarrow (4,5)}{\frac {{\sqrt {x+y}}-3}{x+y-9}}}$

1/6

At what points is the function f continuous?

182. ${\displaystyle f(x,y)=\ln |x-y|}$

${\displaystyle \{(x,y)\mid x\neq y\}}$

183. ${\displaystyle f(x,y)=\displaystyle {\frac {\ln(x^{2}+y^{2})}{x-y+1}}}$

All points (x,y) except for (0,0) and the line y=x+1

Use the two-path test to show that the following limits do not exist. (A path does not have to be a straight line.)

184. ${\displaystyle \displaystyle \lim _{(x,y)\rightarrow (0,0)}{\frac {4xy}{3x^{2}+y^{2}}}}$

The limit is 1 along the line y=x, and −1 along the line y=−x

185. ${\displaystyle \displaystyle \lim _{(x,y)\rightarrow (0,0)}{\frac {y}{\sqrt {x^{2}-y^{2}}}}}$

The limit is 0 along the line y=0, and ${\displaystyle 1/{\sqrt {3}}}$ along the line x=2y

186. ${\displaystyle \displaystyle \lim _{(x,y)\rightarrow (0,0)}{\frac {x^{3}-y^{2}}{x^{3}+y^{2}}}}$

The limit is 1 along the line y=0, and −1 along the line x=0

187. ${\displaystyle \displaystyle \lim _{(x,y)\rightarrow (0,0)}{\frac {x^{2}y^{2}+y^{6}}{x^{3}}}}$

The limit is 0 along any line of the form y=mx, and 2 along the parabola ${\displaystyle x=y^{2}}$

## Partial Derivatives

200. Find ${\displaystyle \partial z/\partial x}$ if ${\displaystyle \displaystyle z(x,y)={\frac {1}{\ln(xy)}}}$

${\displaystyle {\frac {-1}{x(\ln(xy))^{2}}}}$

201. Find all three partial derivatives of the function ${\displaystyle \displaystyle f(x,y,z)=xe^{y^{2}+z}}$

${\displaystyle \displaystyle f_{x}=e^{y^{2}+z},\ f_{y}=2xye^{y^{2}+z},\ f_{z}=f.}$

Find the four second partial derivatives of the following functions.

202. ${\displaystyle f(x,y)=\cos(xy)}$

${\displaystyle f_{xx}=-y^{2}\cos(xy),\ f_{yy}=-x^{2}\cos(xy),\ f_{xy}=f_{yx}=-\sin(xy)-xy\cos(xy).}$

203. ${\displaystyle f(x,y)=xe^{y}}$

${\displaystyle f_{xx}=0,\ f_{yy}=xe^{y},\ f_{xy}=f_{yx}=e^{y}.}$

## Chain Rule

Find ${\displaystyle df/dt.}$

220. ${\displaystyle f(x,y)=x^{2}y-xy^{3},\ x(t)=t^{2},\ y(t)=t^{-2}}$

${\displaystyle 2t+4t^{-5}}$

221. ${\displaystyle f(x,y)={\sqrt {x^{2}+y^{2}}},\ x(t)=\cos(2t),\ y(t)=\sin(2t)}$

0

222. ${\displaystyle \displaystyle f(x,y,z)={\frac {x-y}{y+z}},\ x(t)=t,\ \displaystyle y(t)=2t,\ z(t)=3t}$

0

Find ${\displaystyle f_{s},\ f_{t}.}$

223. ${\displaystyle f(x,y)=\sin(x)\cos(2y),\ x=s+t,\ y=s-t}$

$\displaystyle f_s = \cos(s+t)\cos(2s-2t) - 2\sin(s+t)\sin(2s-2t)\\ f_t = \cos(s+t)\cos(2s-2t) + 2\sin(s+t)\sin(2s-2t)$

224. ${\displaystyle \displaystyle f(x,y,z)={\frac {x-z}{y+z}},\ x(t)=s+t,\ y(t)=st,\ z(t)=s-t}$

$\displaystyle \displaystyle f_s = \frac{-2t(t+1)}{(st+s-t)^2}\\ \displaystyle f_t = \frac{2s}{(st+s-t)^2}$

225. The volume of a pyramid with a square base is ${\displaystyle V={\frac {1}{3}}x^{2}h}$, where x is the side of the square base and h is the height of the pyramid. Suppose that ${\displaystyle \displaystyle x(t)={\frac {t}{t+1}}}$ and ${\displaystyle \displaystyle h(t)={\frac {1}{t+1}}}$ for ${\displaystyle t\geq 0.}$ Find ${\displaystyle V'(t).}$

${\displaystyle \displaystyle {\frac {2t-t^{2}}{3(t+1)^{4}}}}$

## Tangent Planes

Find an equation of a plane tangent to the given surface at the given point(s).

240. ${\displaystyle xy\sin(z)=1,\ (1,2,\pi /6),\ (-1,-2,5\pi /6).}$

${\displaystyle (x-1)+{\frac {1}{2}}(y-2)+{\sqrt {3}}(z-\pi /6)=0,\ (x+1)+{\frac {1}{2}}(y+2)+{\sqrt {3}}(z-5\pi /6)=0}$

241. ${\displaystyle z=x^{2}e^{x-y},\ (2,2,4),\ (-1,-1,1).}$

${\displaystyle -8(x-2)+4(y-2)+z-4=0,\ x+y+z+1=0}$

242. ${\displaystyle z=\tan ^{-1}(x+y),\ (0,0,0).}$

${\displaystyle x+y-z=0}$

243. ${\displaystyle \sin(xyz)=1/2,\ (\pi ,1,1/6).}$

${\displaystyle x+\pi y+6\pi z=3\pi }$

## Maximum And Minimum Problems

Find critical points of the function f. When possible, determine whether each critical point corresponds to a local maximum, a local minimum, or a saddle point.

260. ${\displaystyle f(x,y)=x^{4}+2y^{2}-4xy}$

Local minima at (1,1) and (−1,−1), saddle at (0,0)

261. ${\displaystyle f(x,y)=\tan ^{-1}(xy)}$

262. ${\displaystyle f(x,y)=2xye^{-x^{2}-y^{2}}}$

Saddle at (0,0), local maxima at ${\displaystyle (\pm 1/{\sqrt {2}},\pm 1/{\sqrt {2}}),}$ local minima at ${\displaystyle (\pm 1/{\sqrt {2}},\mp 1/{\sqrt {2}})}$

Find absolute maximum and minimum values of the function f on the set R.

263. ${\displaystyle f(x,y)=x^{2}+y^{2}-2y+1,\ R=\{(x,y)\mid x^{2}+y^{2}\leq 4\}}$

Maximum of 9 at (0,−2) and minimum of 0 at (0,1)

264. ${\displaystyle f(x,y)=x^{2}+y^{2}-2x-2y,}$ R is a closed triangle with vertices (0,0), (2,0), and (0,2).

Maximum of 0 at (2,0), (0,2), and (0,0); minimum of −2 at (1,1)

265. Find the point on the plane xy+z=2 closest to the point (1,1,1).

${\displaystyle (4/3,2/3,4/3)}$

266. Find the point on the surface ${\displaystyle z=x^{2}+y^{2}+10}$ closest to the plane ${\displaystyle x+2y-z=0.}$

${\displaystyle (0.5,1,11.25)}$

## Double Integrals over Rectangular Regions

Evaluate the given integral over the region R.

280. ${\displaystyle \displaystyle \iint _{R}(x^{2}+xy)dA,\ R=\{(x,y)\mid x\in [1,2],\ y\in [-1,1]\}}$

${\displaystyle 14/3}$

281. ${\displaystyle \displaystyle \iint _{R}(xy\sin(x^{2}))dA,\ R=\{(x,y)\mid x\in [0,{\sqrt {\pi /2}}],\ y\in [0,1]\}}$

${\displaystyle 1/4}$

282. ${\displaystyle \displaystyle \iint _{R}{\frac {x}{(1+xy)^{2}}}dA,\ R=\{(x,y)\mid x\in [0,4],\ y\in [1,2]\}}$

${\displaystyle \ln(5/3)}$

Evaluate the given iterated integrals.

283. ${\displaystyle \displaystyle \int _{0}^{2}\int _{0}^{1}x^{5}y^{2}e^{x^{3}y^{3}}dydx}$

${\displaystyle (e^{8}-9)/9}$

284. ${\displaystyle \displaystyle \int _{1}^{4}\int _{0}^{2}e^{y{\sqrt {x}}}dydx}$

${\displaystyle e^{4}-e^{2}-2}$

## Double Integrals over General Regions

Evaluate the following integrals.

300. ${\displaystyle \displaystyle \iint _{R}xydA,}$ R is bounded by x=0, y=2x+1, and y=5−2x.

${\displaystyle 2}$

301. ${\displaystyle \displaystyle \iint _{R}(x+y)dA,}$ R is in the first quadrant and bounded by x=0, ${\displaystyle y=x^{2},}$ and ${\displaystyle y=8-x^{2}.}$

${\displaystyle 152/3}$

Use double integrals to compute the volume of the given region.

302. The solid in the first octant bound by the coordinate planes and the surface ${\displaystyle z=8-x^{2}-2y^{2}.}$

${\displaystyle 4\pi {\sqrt {2}}}$

303. The solid beneath the cylinder ${\displaystyle z=y^{2}}$ and above the region ${\displaystyle R=\{(x,y)\mid y\in [0,1],\ x\in [y,1]\}.}$

${\displaystyle 1/12}$

304. The solid bounded by the paraboloids ${\displaystyle z=x^{2}+y^{2}}$ and ${\displaystyle z=50-x^{2}-y^{2}.}$

${\displaystyle 625\pi }$

## Double Integrals in Polar Coordinates

320. Evaluate ${\displaystyle \displaystyle \iint _{R}2xydA}$ for ${\displaystyle R=\{(r,\theta )\mid r\in [1,3],\ \theta \in [0,\pi /2]\}}$

${\displaystyle 20}$

321. Find the average value of the function ${\displaystyle f(r,\theta )=1/r^{2}}$ over the region ${\displaystyle \{(r,\theta )\mid r\in [2,4]\}.}$

${\displaystyle \displaystyle {\frac {\ln 2}{6}}}$

322. Evaluate ${\displaystyle \displaystyle \int _{0}^{3}\int _{0}^{\sqrt {9-x^{2}}}{\sqrt {x^{2}+y^{2}}}dydx.}$

${\displaystyle 9\pi /2}$

323. Evaluate ${\displaystyle \displaystyle \iint _{R}{\frac {x-y}{x^{2}+y^{2}+1}}dA}$ if R is the unit disk centered at the origin.

${\displaystyle 0}$

## Triple Integrals

340. Evaluate ${\displaystyle \displaystyle \int _{1}^{\ln 8}\int _{0}^{\ln 4}\int _{0}^{\ln 2}e^{-x-y-2z}dxdydz.}$

${\displaystyle \displaystyle {\frac {3}{16}}\left(e^{-2}-8^{-2}\right)}$

In the following exercises, sketching the region of integration may be helpful.

341. Find the volume of the solid in the first octant bounded by the plane 2x+3y+6z=12 and the coordinate planes.

${\displaystyle 8}$

342. Find the volume of the solid in the first octant bounded by the cylinder ${\displaystyle z=\sin(y)}$ for ${\displaystyle y\in [0,\pi ]}$, and the planes y=x and x=0.

${\displaystyle \pi }$

343. Evaluate ${\displaystyle \displaystyle \int _{0}^{1}\int _{y}^{2-y}\int _{0}^{2-x-y}xydzdxdy.}$

${\displaystyle 2/15}$

344. Rewrite the integral ${\displaystyle \displaystyle \int _{0}^{1}\int _{-2}^{2}\int _{0}^{\sqrt {4-y^{2}}}dzdydx}$ in the order dydzdx.

${\displaystyle \displaystyle \int _{0}^{1}\int _{0}^{2}\int _{-{\sqrt {4-z^{2}}}}^{\sqrt {4-z^{2}}}dydzdx}$

## Cylindrical And Spherical Coordinates

360. Evaluate the integral in cylindrical coordinates: ${\displaystyle \displaystyle \int _{0}^{3}\int _{0}^{\sqrt {9-x^{2}}}\int _{0}^{\sqrt {x^{2}+y^{2}}}{\frac {1}{\sqrt {x^{2}+y^{2}}}}dzdydx}$

${\displaystyle 9\pi /4}$

361. Find the mass of the solid cylinder ${\displaystyle D=\{(r,\theta ,z)\mid r\in [0,3],\ z\in [0,2]\}}$ given the density function ${\displaystyle \delta (r,\theta ,z)=5e^{-r^{2}}}$

${\displaystyle 10\pi (1-e^{-9})}$

362. Use a triple integral to find the volume of the region bounded by the plane z=0 and the hyperboloid ${\displaystyle z={\sqrt {17}}-{\sqrt {1+x^{2}+y^{2}}}}$

${\displaystyle \displaystyle {\frac {2\pi (1+7{\sqrt {17}})}{3}}}$

363. If D is a unit ball, use a triple integral in spherical coordinates to evaluate ${\displaystyle \iiint _{D}(x^{2}+y^{2}+z^{2})^{5/2}dV}$

${\displaystyle \pi /2}$

364. Find the mass of a solid cone ${\displaystyle \{(\rho ,\phi ,\theta )\mid \phi \leq \pi /3,\ z\in [0,4]\}}$ if the density function is ${\displaystyle \delta (\rho ,\phi ,\theta )=5-z}$

${\displaystyle 128\pi }$

365. Find the volume of the region common to two cylinders: ${\displaystyle x^{2}+z^{2}=1,\ y^{2}+z^{2}=1}$

${\displaystyle 16/3}$

## Center of Mass and Centroid

380. Find the center of mass for three particles located in space at (1,2,3), (0,0,1), and (1,1,0), with masses 2, 1, and 1 respectively.

${\displaystyle {\frac {\langle 3,5,7\rangle }{4}}}$

381. Find the center of mass for a piece of wire with the density ${\displaystyle \rho (x)=1+\sin(x)}$ for ${\displaystyle x\in [0,\pi ].}$

${\displaystyle \pi /2}$

382. Find the center of mass for a piece of wire with the density ${\displaystyle \rho (x)=2-x^{2}/16}$ for ${\displaystyle x\in [0,4].}$

${\displaystyle 9/5}$

383. Find the centroid of the region in the first quadrant bounded by the coordinate axes and ${\displaystyle x^{2}+y^{2}=16.}$

${\displaystyle \left({\frac {16}{3\pi }},{\frac {16}{3\pi }}\right)}$

384. Find the centroid of the region in the first quadrant bounded by ${\displaystyle y=\ln(x)}$, ${\displaystyle y=0}$, and ${\displaystyle x=e}$.

${\displaystyle ((e^{2}+1)/4,e/2-1)}$

385. Find the center of mass for the region ${\displaystyle \{(x,y)\mid x\in [0,4],y\in [0,2]\}}$, with the density ${\displaystyle \rho (x,y)=1+x/2.}$

${\displaystyle (7/3,1)}$

386. Find the center of mass for the triangular plate with vertices (0,0), (0,4), and (4,0), with density ${\displaystyle \rho (x,y)=1+x+y.}$

${\displaystyle (16/11,16/11)}$

## Vector Fields

One can sketch two-dimensional vector fields by plotting vector values, flow curves, and/or equipotential curves.

401. Find and sketch the gradient field ${\displaystyle \mathbf {F} =\nabla \phi }$ for the potential function ${\displaystyle \phi (x,y)={\sqrt {x^{2}+y^{2}}}}$.

${\displaystyle \mathbf {F} =\left\langle {\frac {x}{\sqrt {x^{2}+y^{2}}}},{\frac {y}{\sqrt {x^{2}+y^{2}}}}\right\rangle }$

402. Find and sketch the gradient field ${\displaystyle \mathbf {F} =\nabla \phi }$ for the potential function ${\displaystyle \phi (x,y)=\sin(x)\sin(y)}$ for ${\displaystyle |x|\leq \pi }$ and ${\displaystyle |y|\leq \pi }$.

${\displaystyle \nabla \phi (x,y)=\langle \cos(x)\sin(y),\sin(x)\cos(y)\rangle }$

403. Find the gradient field ${\displaystyle \mathbf {F} =\nabla \phi }$ for the potential function ${\displaystyle \phi (x,y,z)=e^{-z}\sin(x+y)}$

${\displaystyle \mathbf {F} =e^{-z}\left\langle \cos(x+y),\cos(x+y),-\sin(x+y)\right\rangle }$

## Line Integrals

420. Evaluate ${\displaystyle \int _{C}(x^{2}+y^{2})ds}$ if C is the line segment from (0,0) to (5,5)

${\displaystyle {\frac {250{\sqrt {2}}}{3}}}$

421. Evaluate ${\displaystyle \int _{C}(x^{2}+y^{2})ds}$ if C is the circle of radius 4 centered at the origin

${\displaystyle 128\pi }$

422. Evaluate ${\displaystyle \int _{C}(y-z)ds}$ if C is the helix ${\displaystyle \mathbf {r} (t)=\langle 3\cos(t),3\sin(t),t\rangle ,\ t\in [0,2\pi ]}$

${\displaystyle -2{\sqrt {10}}\pi ^{2}}$

423. Evaluate ${\displaystyle \int _{C}\mathbf {F} \cdot d\mathbf {r} }$ if ${\displaystyle \mathbf {F} =\langle x,y\rangle }$ and C is the arc of the parabola ${\displaystyle \mathbf {r} (t)=\langle 4t,t^{2}\rangle ,\ t\in [0,1]}$

${\displaystyle 17/2}$

424. Find the work required to move an object from (1,1,1) to (8,4,2) along a straight line in the force field ${\displaystyle \displaystyle \mathbf {F} ={\frac {\langle x,y,z\rangle }{x^{2}+y^{2}+z^{2}}}}$

${\displaystyle \ln(2{\sqrt {7}})}$

## Conservative Vector Fields

Determine if the following vector fields are conservative on ${\displaystyle \mathbb {R} ^{2}.}$

440. ${\displaystyle \langle -y,x+y\rangle }$

No

441. ${\displaystyle \langle 2x^{3}+xy^{2},2y^{3}+x^{2}y\rangle }$

Yes

Determine if the following vector fields are conservative on their respective domains in ${\displaystyle \mathbb {R} ^{3}.}$ When possible, find the potential function.

442. ${\displaystyle \langle y,x,1\rangle }$

${\displaystyle \phi (x,y,z)=xy+z}$

443. ${\displaystyle \langle x^{3},2y,-z^{3}\rangle }$

${\displaystyle \phi (x,y,z)=(x^{4}+4y^{2}-z^{4})/4}$

## Green's Theorem

460. Evaluate the circulation of the field ${\displaystyle \mathbf {F} =\langle 2xy,x^{2}-y^{2}\rangle }$ over the boundary of the region above y=0 and below y=x(2-x) in two different ways, and compare the answers.

${\displaystyle 0}$

461. Evaluate the circulation of the field ${\displaystyle \mathbf {F} =\langle 0,x^{2}+y^{2}\rangle }$ over the unit circle centered at the origin in two different ways, and compare the answers.

${\displaystyle 0}$

462. Evaluate the flux of the field ${\displaystyle \mathbf {F} =\langle y,-x\rangle }$ over the square with vertices (0,0), (1,0), (1,1), and (0,1) in two different ways, and compare the answers.

${\displaystyle 0}$

## Divergence And Curl

480. Find the divergence of ${\displaystyle \langle 2x,4y,-3z\rangle }$

${\displaystyle 3}$

481. Find the divergence of ${\displaystyle \displaystyle {\frac {\langle x,y,z\rangle }{1+x^{2}+y^{2}}}}$

${\displaystyle \displaystyle {\frac {x^{2}+y^{2}+3}{(1+x^{2}+y^{2})^{2}}}}$

482. Find the curl of ${\displaystyle \langle x^{2}-y^{2},xy,z\rangle }$

${\displaystyle \langle 0,0,3y\rangle }$

483. Find the curl of ${\displaystyle \langle z^{2}\sin(y),xz^{2}\cos(y),2xz\sin(y)\rangle }$

${\displaystyle \mathbf {0} }$

484. Prove that the general rotation field ${\displaystyle \mathbf {F} =\mathbf {a} \times \mathbf {r} }$, where ${\displaystyle \mathbf {a} }$ is a non-zero constant vector and ${\displaystyle \mathbf {r} =\langle x,y,z\rangle }$, has zero divergence, and the curl of ${\displaystyle \mathbf {F} }$ is ${\displaystyle 2\mathbf {a} }$.

If ${\displaystyle \mathbf {a} =\langle a_{1},a_{2},a_{3}\rangle }$, then

${\displaystyle \mathbf {F} =\mathbf {a} \times \mathbf {r} =\langle a_{2}z-a_{3}y,a_{3}x-a_{1}z,a_{1}y-a_{2}x\rangle =\langle f,g,h\rangle }$, and then

${\displaystyle \nabla \cdot \mathbf {F} =\mathbf {f} _{x}+\mathbf {g} _{y}+\mathbf {h} _{z}=0+0+0=0,}$

${\displaystyle \nabla \times \mathbf {F} =\langle h_{y}-g_{z},f_{z}-h_{x},g_{x}-f_{y}\rangle =\langle a_{1}+a_{1},a_{2}+a_{2},a_{3}+a_{3}\rangle =2\mathbf {a} .}$

## Surface Integrals

500. Give a parametric description of the plane ${\displaystyle 2x-4y+3z=16.}$

${\displaystyle \langle u,v,(16-2u+4v)/3\rangle ,\ u,v\in \mathbb {R} }$

501. Give a parametric description of the hyperboloid ${\displaystyle z^{2}=1+x^{2}+y^{2}.}$

${\displaystyle \langle {\sqrt {v^{2}-1}}\cos(u),{\sqrt {v^{2}-1}}\sin(u),v\rangle ,\ u\in [0,2\pi ],\ |v|\geq 1}$

502. Integrate ${\displaystyle f(x,y,z)=xy}$ over the portion of the plane z=2−xy in the first octant.

${\displaystyle 2/{\sqrt {3}}}$

503. Integrate ${\displaystyle f(x,y,z)=x^{2}+y^{2}}$ over the paraboloid ${\displaystyle z=x^{2}+y^{2},\ z\in [0,4].}$

${\displaystyle {\frac {(391{\sqrt {17}}+1)\pi }{60}}}$

504. Find the flux of the field ${\displaystyle \mathbf {F} =\langle x,y,z\rangle }$ across the surface of the cone
${\displaystyle z^{2}=x^{2}+y^{2},\ z\in [0,1],}$
with normal vectors pointing in the positive z direction.

${\displaystyle 0}$

505. Find the flux of the field ${\displaystyle \mathbf {F} =\langle -y,z,1\rangle }$ across the surface
${\displaystyle y=x^{2},\ z\in [0,4],\ x\in [0,1],}$
with normal vectors pointing in the positive y direction.

${\displaystyle -10}$

## Stokes' Theorem

520. Use a surface integral to evaluate the circulation of the field ${\displaystyle \mathbf {F} =\langle x^{2}-z^{2},y,2xz\rangle }$ on the boundary of the plane ${\displaystyle z=4-x-y}$ in the first octant.

${\displaystyle {\frac {-128}{3}}}$

521. Use a surface integral to evaluate the circulation of the field ${\displaystyle \mathbf {F} =\langle y^{2},-z^{2},x\rangle }$ on the circle ${\displaystyle \mathbf {r} (t)=\langle 3\cos(t),4\cos(t),5\sin(t)\rangle .}$

${\displaystyle 15\pi }$

522. Use a line integral to find ${\displaystyle \iint _{S}(\nabla \times F)\cdot \mathbf {n} dS}$
where ${\displaystyle \mathbf {F} =\langle x,y,z\rangle }$, ${\displaystyle S}$ is the upper half of the ellipsoid ${\displaystyle {\frac {x^{2}}{4}}+{\frac {y^{2}}{9}}+z^{2}=1}$, and ${\displaystyle \mathbf {n} }$ points in the direction of the z-axis.

${\displaystyle 0}$

523. Use a line integral to find ${\displaystyle \iint _{S}(\nabla \times F)\cdot \mathbf {n} dS}$
where ${\displaystyle \mathbf {F} =\langle 2y,-z,x-y-z\rangle }$, ${\displaystyle S}$ is the part of the sphere ${\displaystyle x^{2}+y^{2}+z^{2}=25}$ for ${\displaystyle 3\leq z\leq 5}$, and ${\displaystyle \mathbf {n} }$ points in the direction of the z-axis.

${\displaystyle -32\pi }$

## Divergence Theorem

Compute the net outward flux of the given field across the given surface.

540. ${\displaystyle \mathbf {F} =\langle x,-2y,3z\rangle }$, ${\displaystyle S}$ is a sphere of radius ${\displaystyle {\sqrt {6}}}$ centered at the origin.

${\displaystyle 16{\sqrt {6}}\pi }$

541. ${\displaystyle \mathbf {F} =\langle x,2y,z\rangle }$, ${\displaystyle S}$ is the boundary of the tetrahedron in the first octant bounded by ${\displaystyle x+y+z=1}$

${\displaystyle 2/3}$

542. ${\displaystyle \mathbf {F} =\langle y+z,x+z,x+y\rangle }$, ${\displaystyle S}$ is the boundary of the cube ${\displaystyle \{(x,y,z)\mid |x|\leq 1,|y|\leq 1,|z|\leq 1\}}$

${\displaystyle 0}$

543. ${\displaystyle \mathbf {F} =\langle x,y,z\rangle }$, ${\displaystyle S}$ is the surface of the region bounded by the paraboloid ${\displaystyle z=4-x^{2}-y^{2}}$ and the xy-plane.

${\displaystyle 24\pi }$

544. ${\displaystyle \mathbf {F} =\langle z-x,x-y,2y-z\rangle }$, ${\displaystyle S}$ is the boundary of the region between the concentric spheres of radii 2 and 4, centered at the origin.

${\displaystyle -224\pi }$

545. ${\displaystyle \mathbf {F} =\langle x,2y,3z\rangle }$, ${\displaystyle S}$ is the boundary of the region between the cylinders ${\displaystyle x^{2}+y^{2}=1}$ and ${\displaystyle x^{2}+y^{2}=4}$ and cut off by planes ${\displaystyle z=0}$ and ${\displaystyle z=8}$

${\displaystyle 144\pi }$