# Math 53 Midterm Review

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
pi/3 + sqrt(3)/2
(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
4pi/3
(15.4: 21)
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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)
12 arcsin(2/3)
(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)
1/60
(15.6: 15)
Write the equations in cylindrical coordinates.
z = x^2 – y^2
z = r^2 cos2theta
(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
0
(15.8: 29)
Write the equation in spherical coordinates
z^2 = x^2 + y^2
cos(2Φ)=0
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)
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.
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)
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)
Find the gradient vector field of f.
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
1638.4
(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)
(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
(6/5) – cos(1) – sin(1)
(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)
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) = 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)
a) xyz + z^2
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)
d(2y^3/2)/dy = d(3xsqrt(y))/dx
30
Use a double integral to find the area of the region.
One loop of the rose r = cos3theta
pi/12
(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)
-16/3
(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)
Find the area under one arc of the cycloid x = t-sint, y = 1-cost
3pi
(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)
3x – y +3z = 3
(16.6: 33)
Find the area of the surface.
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
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_
0
(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.
-4pi/3
(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
0
(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
80pi
(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.
3
(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
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
32pi/3
(15.9: 11)
Find the exact length of the curve.
x = 1+3t^2, y = 4+2t^3, 0_
2(2sqrt(2) – 1)
(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)
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
sqrt(3)
(10.3: 55)
Find the exact length of the polar curve.
r = 2cos(theta), 0 _< theta _< pi
2pi
(10.4: 45)
Find the area of the region enclosed by one loop of the curve.
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)
(a) <13, -14, 5>
(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)
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
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
x=1+t,y=-1+2t,z=1+t
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>
x-2y+ 5z= 0
(12.5: 23)
Find an equation of the plane.
The plane through the points (0,1,1), (1,0,1) and (1,1,0)
x+y+z=2
(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)
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
(3/5)j + (4/5)k
(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)
find the length of the curve.
r(t) = sqrt(2) t i + e^t j + 3^-t k
0 _< t _< 1
e – e^(-1)
(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)
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)
does not exist
(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)
x+y-2z=0
(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)
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)
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)
Find the differential of the function.
R = aB^2 cos(r)
dR = B^2 cos®da + 2aBcos®dB – aB^2 sin®dr
(14.4: 29)
Find dz/dx.
x^2 + 2y^2 + 3z^2 = 1
dz/dx = -(x/3z)
(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)
sqrt(3) – (3/2)
(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
-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)
a) x+y+z= 11
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