9.34. LetH = [−1,1]×[0,1], and f(x, y) =
(|x| ify∈Q 0 ify /∈Q
Show that
1
Z
0
1
Z
−1
f(x, y)dx
dy = 0, and f is not integrable on H! 9.35. LetH ={x2+y2≤1}, and
f(x, y) =
(1 ifx≥0
−1 ifx <0
Calculate the lower and upper integral offon the setH! Isf integrable on the setH?
Are the following functions integrable on the unit square N = [0,1]×[0,1]? If the answer is yes, calculate the values of the inte-grals!
9.36. f(x, y) =
(0 ify > x 1 ify≤x 9.37. f(x, y) =
(1 ify≥x 0 ify < x 9.38. f(x, y) =
(0 ifxy6= 0 1 ifxy= 0 9.39. f(x, y) =
(1 ifx, y∈Q 0 otherwise
9.40. f(x, y) =
is the Dirichlet func-tion.
9.43. Letf(x, y) =
n ifx+y= 1/n, n∈N+
0 otherwise .
Show thatf is not integrable on theN unit square, but
1
Calculate the following integrals on the setN = [0,1]×[0,1]! Apply Fubini’s theorem!
9.54. f(x, y) =
Find the following integrals on the given rectangles:
9.58.
Find the following integrals on the given sets:
9.68.
Find the area of the domains bounded by the following curves!
9.78. y=x2, x=y2 9.79. y= 2x−x2, y=x2 9.80. 2y =x2, y=x
9.81. 4y =x2−4x, x−y−3 = 0 9.82. y=x2, y= 2x2, xy= 1, xy= 2
9.83. x2−y2= 1, x2−y2= 4, xy= 1, xy= 2
9.84. Find the area of the setH={(x, y) :−1≤x≤1,0≤y≤p
1−x2}.
9.85. Calculate the area of the two-dimensional shape bounded by the parabo-lasy =x2, y= 2x2, and the linex= 1.
9.86. Calculate the volume of the solid body bounded by the cylinder lateral x2 + y2 = 1, and the planes x+y+z = 2, z= 0. The sought solid body:
9.87. Calculate the volume of the solid body bounded by the cylinder x2+y2 = 1, and the planes x+y+z = 1, z = 0.
The sought solid body:
9.88. Calculate the volume of the solid body below the function f(x, y) = 1−x2
2 −y2
2 and above the setH, ifH = [0,1]×[0,1]!
9.89. Calculate the volume of the solid body below the graph of the function f(x, y) =x+yand above the setH, ifH ={0≤x+y≤1,0≤x,0≤ y}!
Plot the solid body whose volume can be calculated by the following integrals. Calculate the integrals!
9.90.
Calculate the volume of the solid bodies bounded by the following surfaces!
Calculate the volumes of the following solid bodies!
9.96. sphere 9.97. ellipsoid
9.98. circular cylinder 9.99. circular cone
Let us assume that the two-dimensional range H consist of some material with density %(x, y). The mass of the shape:
M = Z Z
H
%(x, y)dx dy,
the coordinates of the center of mass are Sx = 1
9.100. %(x, y) =x2 9.101. %(x, y) =x+y 9.102. %(x, y) =xy 9.103. %(x, y) =x2+y2
Find the center of mass of the two-dimensional shape which is bounded by the linesy= 0, x= 2,y= 1, y =x, and its density is 9.104. %(x, y) = 1 9.105. %(x, y) =x
9.106. %(x, y) =y 9.107. %(x, y) =xy 9.108. %(x, y) = 1
x+y3 9.109. %(x, y) =ex+y
The moment of inertia of a rigid body on the planexywith respect to z axis is
Θ = Z Z
H
r2(x, y)%(x, y)dx dy,
where r(x, y) is the distance of the point (x, y) from the z axis.
Find the moment of inertia of the unit square on thexyplane with density % with respect to the z axis if one vertex of the square is on the z axis.
9.110. %(x, y) = 1 9.111. %(x, y) =xy
9.112. Find the moment of inertia of the squares in the previous problems with respect to thezaxis if the midpoint of one side of the square is on the z axis!
9.113. A thin membrane is bounded by the liney= 0, x= 1 andy= 2x. The density of the membrane is%(x, y) = 6x+ 6y+ 6. Find the mass of the membrane and coordinates of the center of mass!
9.114. A rigid body is in the first octant, it is bounded by the coordinate planes and the planex+y+z= 2, and its density is %(x, y, z) = 2x.
Find the mass of the membrane and the coordinates of the center of mass!
9.115. We pump the water to the surface from a sump whose depth is 1 meter.
What amount of work is needed against the gravitation if the sump is
(a) a cube (b) a half sphere?
Line Integral and Primitive Function
10.1 Tangent line. The equation of the spatial curve r(t) at the point r0=r(t0) is
r =r0+v ·t,
wherev = ˙r(t0) is the direction vector of the tangent line.
10.2 Arc length for planar and spatial curves.
— If the planar curver : [a, b]→R2is continuously differentiable, then it is rectifiable, and the length of its arc is
L=
b
Z
a
|r˙| dt=
b
Z
a
px˙2+ ˙y2dt.
— Iff : [a, b]→Ris continuously differentiable, then its graph is rectifi-able, and the length of its arc is
L=
b
Z
a
p1 + (f0)2dx.
— If the spatial curver : [a, b]→R3 is continuously differentiable, then it is rectifiable, and the length of its arc is
L=
b
Z
a
|r˙|dt=
b
Z
a
px˙2+ ˙y2+ ˙z2dt.
10.3 Tangent plane. The equation of the tangent plane of the surface r(u, v) at the pointr0=r(u0, v0) is
n·r =n·r0,
wheren =r0u(u0, v0)×r0v(u0, v0) is the normal vector to the tangent plane.
Especially, the tangent plane of the graph ofz=f(x, y) at the point (x0, y0) is
z=f(x0, y0) +fx0(x0, y0)(x−x0) +fy0(x0, y0)(y−y0).
10.4 Surface area. If the surfacer :A→R3is continuously differentiable, then the (finite) area of the surface exists, and
S= Z Z
A
|r0u×r0v| du dv.
Especially, the surface of the graph of the continuously differentiable function z=f(x, y) over the measurable planar regionAis
S= Z Z
A
q
1 + (zx0)2+ (zy0)2dx dy.
10.5 Calculating the line integral. If the vector fieldv =v1(x, y, z)·i+ v2(x, y, z)·j +v3(x, y, z)·k is continuous on the regionG, andr : [a, b]→ G, r(t) =x(t)·i+y(t)·j +z(t)·k is continuously differentiable, then the line integral ofv exists along the curve Γ :r(t), and
Z
Γ
v dr =
b
Z
a
v(r(t))·r˙(t)dt.
With coordinates Z
Γ
v1(x, y, z)dx+v2(x, y, z)dy+v3(x, y, z)dz=
=
b
Z
a
[v1(x, y, z) ˙x+v2(x, y, z) ˙y+v3(x, y, z) ˙z] dt.
In the case of planar vector field and curve Z
Γ
v1(x, y)dx+v2(x, y)dy=
b
Z
a
[v1(x, y) ˙x+v2(x, y) ˙y]dt.
10.6 Conservative vector field.
— The vector fieldv is conservative on the regionGif and only if along all closed and rectifiable Γ curves insideG
I
Γ
vdr = 0,
that is, all closed line integrals are zero.
— Newton-Leibniz formulafor line integrals.
If the vector fieldv is conservative on the regionG,U(r) is a primitive function of v onG, and Γ is a continuously differentiable curve inG with starting pointa and endpoint b, then
Z
Γ
vdr =U(b)−U(a).
— Ifv is a continuously differentiable and conservative vector field on the regionG, then
curlv =0, that is, the vector field isirrotational.
— If the vector fieldv is continuously differentiable on the simply con-necteddomainG, and at the point ofGcurlv =0, thenv is conser-vative.
10.1 Planar and Spatial Curves
Plot the following planar curves!
10.1. r =t·i+t2·j t∈[0,4]
10.2. r =t2·i +t·j t∈[0,16]
10.3. r =√
t·i +t·j t∈[0,16]
10.4. r =t·i +√ t·j t∈[0,4]
10.5. r = 2t·i + 4t2·j t∈[0,2]
10.6. r =t2·i +t2·j t∈[0,4]
10.7. r = cost·i + sint·j t∈[0,2π]
10.8. r = cost·i+ sint·j t∈[0, π]
10.9. r = cost·i + sint·j t∈[−π/2, π/2]
10.10. r = 2 cost·i+ 4 sint·j t∈[0,2π]
10.11. r = 4 cost·i + 2 sint·j t∈[π/2,3π/2]
10.12. r = cost·i+tsint·j t∈[0,2π]
10.13. r =tcost·i + sint·j t∈[0,6π]
10.14. r =tcost·i+tsint·j t∈[0,4π]
Plot the following spatial curves!
10.15. r =t·i+ 2t·j + 3t·k t∈[2,4]
10.16. r =−2t·i+t·j −(t/3)·k t∈[2,4]
10.17. r = cost·i + sint·j +t·k t∈[0,6π]
10.18. r =t·i+ sint·j + cost·k t∈[0,6π]
10.19. r = 2 sint·i−t2·j + cost·k t∈[2,6π]
10.20. r =tcost·i +tsint·j +t·k t∈[2,6π]
10.21. Give an example of a curve which is a (a) cylindrical spiral;
(b) conical spiral!
10.22. Find the arc length of the previous curves if the height and radius of the cylinder, the circle at the bottom of the cone are given!
Plot the following curves! Write down the equation of the tangent lines at t=π/4!
10.23. r(t) = 2 cost·i + 3 sint·j t∈[0,8π]
10.24. r(t) =tcost·i +tsint·j t∈[0,8π]
Write down the equation of the following planar curves at the given points!
10.25. x2−xy3+y5= 17 P(5,2) 10.26. (x2+y2)2= 3x2y−y3 P(0,0)
Calculate the equations of the tangent lines of the following spatial curves at the given points!
10.27. r(t) = (t−3)·i+(t2+1)·j+t2·k t= 2 10.28. r(t) = sint·i+ cost·j+ 1
cost·k p =j +k
Find the arc length of the following planar curves!
10.29. (cykloid) x=r(t−sint) y=r(1−cost)
0≤t≤2π
10.30. (Archimedean spiral)
r=aϕ 0≤ϕ≤2π
10.31. y=√
x 0≤x≤a