Practice problems
1. Evaluate the double or iterated integrals:
RR √
√
• R x3 + 1dA where R = {(x, y) : 0 ≤ y ≤ 1, y ≤ x ≤ 1}.
R 1/2 R √1−y2
√
• 0
sin(x2 + y 2 )dxdy
3y
First: change the order of integration; Second: polar.
2. Consider the ice-cream cone shaped lamina bounded by x2 + y 2 =
1, y ≥ 0 and y = |x| − 1, |x| ≤ 1. The density is δ = 1. Set up
the integral for the total mass (which equals the area) in Cartesian
coordinates using the order dxdy. (Hint: You need to break the region
into two parts.) If I ask you to evaluate the integrals, will you change
the coordinates for one of the regions?
3. Set up the integral for the volume inside both x2 +y 2 = 1 and x2 +y 2 −
2y = 0, below z = 2y 2 + x2 and above xy plane. Rewrite it in polar
coordinates. If this region has a mass density δ = x2 + 2z 2 , set up the
integral for the moment of inertia about y axis (don’t evaluate).
4. Evaluate the volume bounded by the paraboloid z = x2 + y 2 and the
plane z = y.
This is the problem that involves the polar coordinates. In double
integral it is
ZZ
(y − x2 − y 2 )dA
R
where R is the region bounded by x2 + y 2 = y. Then, apply polar.
This is one example in the lecture notes.
This problem can also be evaluated using triple integral by cylindrical
coordinates. It’s essentially the same as the double integral one.
5. Compute the volume of the solid bounded by z = −1, z = y + 1,
z = 1 − y, x = y 2 + z 2 and yz plane.
Consider the three surfaces that have no x variable first. These three
guys form a triangular region in yz plane. For the x direction, it’s
clearly between x = 0 and x = y 2 + z 2 . You can set it up in triple
integral using Cartesian
1
2
2
6. Let T be the region bounded
RRR by y = x , x = y , z = 0 and z = x + y.
Find the triple integral
T xydV .
√
The key is to write out the region: 0 ≤ x ≤ 1, x2 ≤ y ≤ x and
0 ≤ z ≤ x + y.
7. Let R be the region bounded by 2x + y = 3, y − 2x = 2, 2x + y =
1, 2x − y = −4. Compute the double integral
ZZ
(16x2 − 4y 2 )dxdy.
R
We use change of variables u = 2x + y, v = y − 2x. Then, we compute
the Jacobian.
8. Let R be the parallelogram with vertices (0, 0), (1, 1), (2, 0), (1, −1).
Find the integral
ZZ
(x2 − y 2 )dA
R
9. Find the area of the region R inside theq
ellipse x2 − xy + y 2 = q
2. (Hint:
√
√
try the change of variables x = 2u − 23 v and y = 2u + 23 v)
10. Let R be the region bounded by yex − 2 = 0, yex − 4 = 0, x3 − 6y − 1 =
0, x3 − 6y − 2 = 0 in the first quadrant. Evaluate the double integral
ZZ
I=
(2ex y + x2 ex )ex y dA.
R
2 +y 2 /4+z 2 /9 ≤ 1. Evaluate the average
11. Consider that T is given by xRRR
1
2
value of f over T : V olume(T )
T f (x, y, z)dV , where f = x .
Do change of variables first u = x, v = y/2, w = z/3. Then, the region
is transformed into a ball.
12. Find the volume under f (x, y) = x and above the region R = {(x, y) :
(x − 1)2 + y 2 ≤ 1, x2 + (y − 1)2 ≤ 1}.
convenient in cylindrical coordinates.
13. Considerpthe ball x2 +y 2 +z 2 ≤ 4. If the density inside x2 +y 2 +z 2 = 1
is δ1 = x2 + y 2 + z 2 while the density outside it is δ = 1. What is
the total mass of the ball?
2
14. Consider the the solid outside x2 + y 2 + (z − 1)2 = 1 but inside ρ =
4 cos φ. Suppose the density is given by δ = x2 + y 2 . Set up integrals
for the centroid and moment of inertia about z axis.
Clearly, use spherical coordinates to set up the triple integral. Then,
δ = ρ2 sin2 φ.
RRR
2
2
15. Evaluate
T xydV where T is the region bounded by x +y −2x = 0
2
2
2
and x + y + z = 4. What is the volume of this region?
Cylindrical coordinates are convenient. The sphere becomes r2 + z 2 =
4.
RRR
2
2
16. Set up the integral for
T f (x, y, z)dV where f = x + y and T is
the region
contained in the sphere x2 + y 2 + (z − a)2 = a2 but below
√
z = 33 r.
Spherical coordinates. T : 0 ≤ θ < 2π, π3 ≤ φ ≤ π/2, 0 ≤ ρ ≤ 2a cos φ.
f = ρ2 sin2 φ.
Surface areas:
• Parametrize the surface y = f (x, z). Use this to compute the area of
the plane y = 2x + 2z + 1 inside x2 + z 2 = 1.
• Set up an integral for the surface area of the surface cut from x =
y 2 + 3z 2 − 2z by z + 2y = 3, z = y 2 .
• Consider the surface of revolution obtained by revolving x = f (z)
about z axis. Parametrize this surface.
• Consider the fence S: x = 2 sin(t), y = 8RR
cos(3t), 0 ≤ t < 2π and
0 ≤ z ≤ 2. Set up the surface area integral S dS.
• S is the surface z = θ, 0 ≤ θ ≤ π and 1 ≤ x2 + y 2 ≤ 4. Set up an
iterated integral for the surface area.
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1. Divergence, curl
(a) In the daytime under sunshine, the algae in an ocean generate
oxygen (they also consume oxygen but the net effect is oxygen
production). The rate of production is clearly proportional to the
density of algae in the ocean. Suppose that the oxygen consumption by other plants and animals in the ocean can be neglected
3
and that the density distribution of the oxygen reaches equilibrium. The equilibrium is kept under a flow of the oxygen which
is the result of diffusion, transportation etc. We model the ocean
by R3 and the field of the oxygen flow is given by
2
F = harctan(x)+e−y ,
x2
sin4 (x)
+arctan(y),
arctan(z)+
i.
1 + z 2 + x4
y4 + 1
If the density of the algae at (0, 0, 0) is 3 × 103 per cubic centimeter, what is the density at (1, 1, 1)?
(b) Suppose there is a cloud of charged dust. The electronic field
generated by the dust is given by
E = hxy 2 , xyz, sin(x)i.
What is the charge density at (0, 0, 0)? If the dust is positively
charged at (1, 1, 1) with 3×10−6 C/cm3 , what is the charge density
at (−1, −1, 2)? Is it negatively charged or positively charged?
(c) In a storm weather, near (π/2, 1, 1), the velocity field of the air
was roughly given by
v = hxy 2 , xyz, sin(x)i.
What is the vorticity at (π/2, 1, 1)?
2. Line integrals
(a) Parametrization
• Parametrize x2 + 4y 2 = 1 r(t) = hcos t, 12 sin ti, 0 ≤ t < 2π
• Parametrize the boundary of the region bounded by x-axis,
√
y = x and x = 1. C = C1 +C2 +C3 . C1 : r(t)√= ht, 0i, 0 ≤
t ≤ 1. C2 : r(t) = h1, ti, 0 ≤ t ≤ 1. C3 : r = ht, ti, t : 1 → 0
• Parametrize the ellipse formed by the intersection of x2 +y 2 =
1 and x + z = 0.
r(t) = hcos t, sin t, − cos ti
(b) Usual line integrals (2 types)
• Consider the curve x2 /4 + y 2 = 1 with x ≥ 0, 0 ≤ y ≤ 1/2. If
the density (per unit length) is δ = y/x, compute the moment
of inertia
R 2Iy .
R
Iy = C x δds = C xyds. r = h2 cos t, sin ti. 0 ≤ t ≤ π/6
4
• Compute the line integral of F = h3y, −2xi over the curve
y = x2 for 0 ≤ y ≤ 1 oriented from right to left.
R
• Let r = ht3 , t2 , ti, 0 ≤ t ≤ 1. Compute C F · T ds where
F = heyz , 0, yeyz i
(c) Conservative field.
9
3
100
• Let
R C be r(t) = hln(1 + t ), t + 1, t i, 0 ≤ t ≤ 1. Compute
C xdy + ydx + dz
R
The field is ∇(xy +z). Or you can notice that it is d(xy +z)
R
• C F · T ds where F = hzexz + ex , 2yz, xexz + y 2 i. r =
2
3
het , et , t4 i. t ∈ [0, 1]
It is irrotational and thus conservative. φ = exz + y 2 z
3
• Let C be ~r(t) = hcos4 t, sin4 t, 7i, 0 ≤ t < 2π
R and F = hx −
3
3
z, y , y + z i. Compute the line integral C F · d~r. (Hint:
Split out a conservative field. The integral of the one you
split will be zero since it is on a closed curve.)
The field F is not conservative but hx3 , y 3 , z 3 i is conservaR is closed, we onlyR need to compute
Rtive. Since the curve
h−z,
0,
yi
·
dr
=
C
C −7dx + ydz = −7 C dx = 0 because
h1, 0, 0i is again conservative.
5