Use a double integral to find the area of the region.

The region inside the circle (x – 1)^2 + y^2 = 1 and outside the circle x^2 + y^2 = 1

The region inside the circle (x – 1)^2 + y^2 = 1 and outside the circle x^2 + y^2 = 1

pi/3 + sqrt(3)/2

(15.4: 17)

(15.4: 17)

Use polar coordinates to find the volume of the given solid.

Enclosed by the hyperboloid -x^2 -y^2 +z^2 = 1 and the plane z = 2

Enclosed by the hyperboloid -x^2 -y^2 +z^2 = 1 and the plane z = 2

4pi/3

(15.4: 21)

(15.4: 21)

Find the area of the surface.

The part of the cylinder y^2 + z^2 = 9 that lies above the rectangle with vertices (0,0), (4,0), (0,2) and (4,2)

The part of the cylinder y^2 + z^2 = 9 that lies above the rectangle with vertices (0,0), (4,0), (0,2) and (4,2)

12 arcsin(2/3)

(15.6: 5)

(15.6: 5)

Evaluate the triple integral

SSS(T) x^2 dv, where T is the solid tetrahedron with vertices (0,0,0), (1,0,0), (0,1,0) and (0,0,1)

SSS(T) x^2 dv, where T is the solid tetrahedron with vertices (0,0,0), (1,0,0), (0,1,0) and (0,0,1)

1/60

(15.6: 15)

(15.6: 15)

Write the equations in cylindrical coordinates.

z = x^2 – y^2

z = x^2 – y^2

z = r^2 cos2theta

(15.8: 9)

(15.8: 9)

Solve:

S(-2 to 2) S(-sqrt(4-y^2) to sqrt(4-y^2) S(sqrt(x^2 + y^2) to 2) xz dzdxdy

S(-2 to 2) S(-sqrt(4-y^2) to sqrt(4-y^2) S(sqrt(x^2 + y^2) to 2) xz dzdxdy

0

(15.8: 29)

(15.8: 29)

Write the equation in spherical coordinates

z^2 = x^2 + y^2

z^2 = x^2 + y^2

cos(2Φ)=0

Φ = pi/4 and 3pi/4

Φ = pi/4 and 3pi/4

Evaluate SSS(E) (x^2 + y^2) dV, where E lies between the spheres x^2 + y^2 + z^2 =4 and x^2 + y^2 + z^2 =9.

1688pi/15

(15.9: 23)

(15.9: 23)

A region R in the xy-plane is given. Find equations for a transformation T that maps a rectangular region S in the uv-plane onto R, where the sides of S are parallel to the u- and v-axes.

R lies between the circles x^2 + y^2 = 1 and x^2 + y^2 = 2 in the first quadrant.

R lies between the circles x^2 + y^2 = 1 and x^2 + y^2 = 2 in the first quadrant.

T is defined by x=ucosv,y=usinv and T maps the rectangle S={(u,v)I1<_u<_sqrt(2), 0<_v<_pi/2} in the uv-plane to R in the xy-plane.
(15.10: 13)

SS® (x-3y)dA, where R is the triangular region with vertices (0,0), (2,1) and (1,2); x = 2u+v, y=u+2v.

-3

(15.10: 15)

(15.10: 15)

SS® cos((y-x)/(y+x))dA, where R is the trapezoidal region with vertices (1,0), (2,0), (0,2), and (0,1)

(3/2)sin(1)

(15.10: 25)

(15.10: 25)

Find the gradient vector field of f.

f(x,y) = xe^(xy)

f(x,y) = xe^(xy)

(xy+1)e^(xy)i + x^2e^(xy)j

Evaluate the line integral, where C is the given curve.

S© xy^4 ds, C is the right half of the circle x^2 + y^2 = 16

S© xy^4 ds, C is the right half of the circle x^2 + y^2 = 16

1638.4

(16.2: 3)

(16.2: 3)

Evaluate the line integral, where C is the given curve.

S© xe^(yz) ds, C is the line segment from (0,0,0) to (1,2,3)

S© xe^(yz) ds, C is the line segment from (0,0,0) to (1,2,3)

(sqrt(14)/12)(e^6 – 1)

Evaluate the line integral S© F*dr, where C is given by the vector function r(t).

F(x,y,z) = sinx i + cosy j + kz k, r(t) = t^3 i – t^2 j + t k, 0 <_ t <_ 1

F(x,y,z) = sinx i + cosy j + kz k, r(t) = t^3 i – t^2 j + t k, 0 <_ t <_ 1

(6/5) – cos(1) – sin(1)

(16.2: 21)

(16.2: 21)

A thin wire is bent into the shape of a semicircle x^2 + y^2 = 4, x >_ 0. If the linear density is a constant k, find the mass and center of mass of the wire.

m = 2k(pi)

center of mass: (4/pi, 0)

center of mass: (4/pi, 0)

Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ▽f.

F(x,y) = (lny + 2xy^3)i + (3x^2y^2 + x/y)j

F(x,y) = (lny + 2xy^3)i + (3x^2y^2 + x/y)j

f(x,y) = xlny + x^2y^3 + K

(a) Find a function f such that F = ▽f and (b) use part (a) to evaluate S© F*dr along the given curve C.

F(x,y,z) = yz i + xz j + (xy + 2z) k, C is the line segment from (1,0,-2) to (4,6,3)

F(x,y,z) = yz i + xz j + (xy + 2z) k, C is the line segment from (1,0,-2) to (4,6,3)

a) xyz + z^2

b) 77

(16.3: 15)

b) 77

(16.3: 15)

Find the work done by the force field F in moving an object from P to Q.

F(x,y) = 2y^3/2 i + 3xy^1/2 j; P(1,1), Q(2,4)

F(x,y) = 2y^3/2 i + 3xy^1/2 j; P(1,1), Q(2,4)

d(2y^3/2)/dy = d(3xsqrt(y))/dx

30

30

Use a double integral to find the area of the region.

One loop of the rose r = cos3theta

One loop of the rose r = cos3theta

pi/12

(15.4: 15)

(15.4: 15)

Use Green’s Theorem to evaluate S© F*dr

F(x,y) =

C is the triangle from (0,0) to (0,4) to (2,0) to (0,0)

F(x,y) =

C is the triangle from (0,0) to (0,4) to (2,0) to (0,0)

-16/3

(16.4: 11)

(16.4: 11)

Fine the work done by the force F(x,y) = x(x+y)i + xy^2j in moving a particle from the origin along the x-axis to (1,0), then along the line segment to (0,1), and then back to the origin along the y-axis.

-1/12

(16.4: 17)

(16.4: 17)

Find the area under one arc of the cycloid x = t-sint, y = 1-cost

3pi

(16.4: 19)

(16.4: 19)

Find an equation of the tangent plane to the given parametric surface at the specified point.

x = u + v, y = 3u^2, z = u – v; (2,3,0)

x = u + v, y = 3u^2, z = u – v; (2,3,0)

3x – y +3z = 3

(16.6: 33)

(16.6: 33)

Find the area of the surface.

The surface z = (2/3)(x^(3/2) + y^(3/2)), 0_

The surface z = (2/3)(x^(3/2) + y^(3/2)), 0_

(4/15)(3^(5/3) – 2^(7/2) + 1)

Set up the surface integral.

SS(s) y dS,

S is the part of paraboloid y = x^2 + z^2 that lies inside the cylinder x^2 + z^2 = 4

SS(s) y dS,

S is the part of paraboloid y = x^2 + z^2 that lies inside the cylinder x^2 + z^2 = 4

S(0 to 2pi)S(0 to 2) r^3 sqrt(4r^2 + 1) dr dtheta

Evaluate the surface integral SS(s) F*ds for the given vector field F and the oriented surface S.

F(x,y,z) = yj – zk

S consists of the paraboloid y = x^2 + z^2, 0_

F(x,y,z) = yj – zk

S consists of the paraboloid y = x^2 + z^2, 0_

0

(16.7: 27)

(16.7: 27)

Find the flux of F across S.

F(x,y,z) = xi – zj + yk,

S is the part of the sphere x^2 + y^2 + z^2 = 4 in the first octant, with orientation toward the origin.

F(x,y,z) = xi – zj + yk,

S is the part of the sphere x^2 + y^2 + z^2 = 4 in the first octant, with orientation toward the origin.

-4pi/3

(16.7: 25)

(16.7: 25)

Use Stoke’s Theorem to evaluate SS(s) curl F*dS

F(x,y,z) = x^2 z^2 i + y^2 z^2 j + xyz k,

S is the part of the paraboloid z = x^2 + y^2 that lies inside cylinder x^2 + y^2 = 4, oriented upward

F(x,y,z) = x^2 z^2 i + y^2 z^2 j + xyz k,

S is the part of the paraboloid z = x^2 + y^2 that lies inside cylinder x^2 + y^2 = 4, oriented upward

0

(15.8: 3)

(15.8: 3)

Use Stoke’s Theorem to evaluate S© F*dr. In each case C is oriented counterclockwise as viewed from above.

F(x,y,z) = yzi + 2xzj + e^(xy)k,

C is the circle x^2 + y^2 = 16, z=5

F(x,y,z) = yzi + 2xzj + e^(xy)k,

C is the circle x^2 + y^2 = 16, z=5

80pi

(15.8: 9)

(15.8: 9)

A particle moves along line segments from the origin to the points (1,0,0), (1,2,1), (0,2,1), and back to the origin under the influence of the force field

F(x,y,z) = z^2 i + 2xy j + 4y^2 k

Find the work done.

F(x,y,z) = z^2 i + 2xy j + 4y^2 k

Find the work done.

3

(15.8: 17)

(15.8: 17)

Vertify that the Divergence Theorem is true for the vector field F on the region E.

F(x,y,z) =,

E is the solid ball x^2 + y^2 + z^2 _< 16

F(x,y,z) =

E is the solid ball x^2 + y^2 + z^2 _< 16

256pi/3

Calculate the flux of F across S.

F(x,y,z) = (cosz + xy^2) i + xe^(-z) j + (siny + x^2 z) k ,

S is the surface of the solid bounded by the paraboloid z = x^2 + y^2 and the plane z = 4

F(x,y,z) = (cosz + xy^2) i + xe^(-z) j + (siny + x^2 z) k ,

S is the surface of the solid bounded by the paraboloid z = x^2 + y^2 and the plane z = 4

32pi/3

(15.9: 11)

(15.9: 11)

Find the exact length of the curve.

x = 1+3t^2, y = 4+2t^3, 0_

x = 1+3t^2, y = 4+2t^3, 0_

2(2sqrt(2) – 1)

(10.2: 41)

(10.2: 41)

Use the parametric equations of an ellipse, x = a cos(theta), y = b sin (theta), 0 _< theta _< 2pi, to find the area that it encloses.

ab pi

(10.2: 31)

(10.2: 31)

Find the slope of the tangent line to the given polar curve at the point specified by the value of theta.

r = 2 sin(theta), theta = pi/6

r = 2 sin(theta), theta = pi/6

sqrt(3)

(10.3: 55)

(10.3: 55)

Find the exact length of the polar curve.

r = 2cos(theta), 0 _< theta _< pi

r = 2cos(theta), 0 _< theta _< pi

2pi

(10.4: 45)

(10.4: 45)

Find the area of the region enclosed by one loop of the curve.

r = 4cos 3(theta)

r = 4cos 3(theta)

(4/3)pi

(a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and (b) find the area of triangle PQR.

P(0,-2,0), Q(4,1,-2), R(5,3,1)

P(0,-2,0), Q(4,1,-2), R(5,3,1)

(a) <13, -14, 5>

(b) (1/2)sqrt(390)

(b) (1/2)sqrt(390)

Find an equation for the plane consisting of all points that are equidistant from the points (1,0,-2) and (3,4,0).

x+2y+z=5

(12.5: 61)

(12.5: 61)

Find a vector equation and parametric equations for the line.

The line through the point (1,0,6) and perpendicular to the plane x + 3y + z = 5

The line through the point (1,0,6) and perpendicular to the plane x + 3y + z = 5

r = (1+t)i+3tj+(6+t)k,

Find parametric equations and symmetric equations for the line.

The line through (1,-1,1) and parallel to the line x + 2 = (1/2)y = z – 3

The line through (1,-1,1) and parallel to the line x + 2 = (1/2)y = z – 3

x=1+t,y=-1+2t,z=1+t

x-1=(y+1)/2=z-1

(12.5: 11)

x-1=(y+1)/2=z-1

(12.5: 11)

Find an equation of the plane.

The plane through the origin and perpendicular to the vector <1, -2, 5>

The plane through the origin and perpendicular to the vector <1, -2, 5>

x-2y+ 5z= 0

(12.5: 23)

(12.5: 23)

Find an equation of the plane.

The plane through the points (0,1,1), (1,0,1) and (1,1,0)

The plane through the points (0,1,1), (1,0,1) and (1,1,0)

x+y+z=2

(12.5: 31)

(12.5: 31)

Find a vector equation and parametric equations for the line segment that joins P to Q.

P(2,0,0), Q(6,2,-2)

P(2,0,0), Q(6,2,-2)

r(t)=(2+4t,2t,-2t); 0_

Find the unit tangent vector at the point with the given value of the parameter t.

r(t) = cost i + 3t j + 2 sin2t k, t = 0

r(t) = cost i + 3t j + 2 sin2t k, t = 0

(3/5)j + (4/5)k

(13.2: 19)

(13.2: 19)

Find f'(2), where f(t) = u(t) * v(t), u(2) = <1,2,-1>, u'(2) = <3,0,4>, and v(t) =

35

(13.2: 49)

(13.2: 49)

find the length of the curve.

r(t) = sqrt(2) t i + e^t j + 3^-t k

0 _< t _< 1

r(t) = sqrt(2) t i + e^t j + 3^-t k

0 _< t _< 1

e – e^(-1)

(13.3: 3)

(13.3: 3)

Let C be the curve of intersection of the parabolic cylinder x^2 = 2y and the surface 3z = xy. Find the exact length C from the origin to the point (6, 18, 36).

42

(13.3: 11)

(13.3: 11)

Find the limit, if it exists, or show that the limit does not exist.

lim(x,y)->(0,0) (y^2 sin^2(x)/(x^4 + y^4)

lim(x,y)->(0,0) (y^2 sin^2(x)/(x^4 + y^4)

does not exist

(Ch. 14.2 11)

(Ch. 14.2 11)

Use implicit differentiation to find dz/dx.

dz/dx = (yz)/(e^z – xy)

Find an equation of the tangent plane to the given surface at the specified point.

z = sqrt(xy), (1,1,1)

z = sqrt(xy), (1,1,1)

x+y-2z=0

(14.4: 3)

(14.4: 3)

Explain why the function is differentiable at the given point. Then find the linearization L(x,y) of the function at that point.

f(x,y) = 1 + xln(xy – 5), (2,3)

f(x,y) = 1 + xln(xy – 5), (2,3)

Both fx and fy are continuous functions for xy>5, so by Theorem 8, f is differentiable at (2, 3).

L(x,y) = 6x + 4y – 23

(ch. 14.4: 11)

L(x,y) = 6x + 4y – 23

(ch. 14.4: 11)

Given that f is differentiable function of f(2,5) = 6, fx(2,5) = 1, and fy(2,5) = -1, use a linear approximation to estimate f(2.2, 4.9)

6.3

(ch. 14.4: 19)

(ch. 14.4: 19)

Find the differential of the function.

R = aB^2 cos(r)

R = aB^2 cos(r)

dR = B^2 cos®da + 2aBcos®dB – aB^2 sin®dr

(14.4: 29)

(14.4: 29)

Find dz/dx.

x^2 + 2y^2 + 3z^2 = 1

x^2 + 2y^2 + 3z^2 = 1

dz/dx = -(x/3z)

(14.5: 31)

(14.5: 31)

Find the rate of change of f at P in the direction of the vector u.

f(x,y) = sin(2x + 3y), P(-6,4), u=(1/2)(sqrt(3)i – j)

f(x,y) = sin(2x + 3y), P(-6,4), u=(1/2)(sqrt(3)i – j)

sqrt(3) – (3/2)

(14.6: 7)

(14.6: 7)

Find the directional derivative of the function at the given point in the direction of the vector v.

g(p,q) = p^4 – p^2 q^3, (2,1)

v = i + 3j

g(p,q) = p^4 – p^2 q^3, (2,1)

v = i + 3j

-8/sqrt(10)

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specific point.

2(x-2)^2 + (y-1)^2 + (z-3)^2 = 10,

(3,3,5)

2(x-2)^2 + (y-1)^2 + (z-3)^2 = 10,

(3,3,5)

a) x+y+z= 11

b) (x-3)/4 = (y-3)/4 = (z-5)/4

b) (x-3)/4 = (y-3)/4 = (z-5)/4

Find the local maximum and minimum values and saddle point(s) of the function.

f(x,y) = x^3 -12xy + 8y^3

f(x,y) = x^3 -12xy + 8y^3

(0,0) is a saddle point

(2,1)= -8 is a local minimum

(2,1)= -8 is a local minimum

Find the volume of the given solid.

Bounded by the coordinate planes and the plane

3x+2y+z=6

Bounded by the coordinate planes and the plane

3x+2y+z=6

6

(Ch. 15.3: 27)

(Ch. 15.3: 27)