MA22S1: Multivariable Calculus for Science
Homework 9, due on December 12
Multiple integrals
1. Sketch the region of integration and evaluate the iterated integral by choosing a convenient order of integration:
Z 2Z 1
cos(y 2 )dydx .
x
2
0
√
2. For a region R enclosed between the graphs of y = x and y = x2 /3 fill
in the missing limits of integration below, i.e. express the double integral
as an iterated integral in two possible ways:
Z Z
Z
Z
f (x, y)dA =
Z
Z
F (x, y)dxdy =
R
F (x, y)dydx .
Sketch the region R and calculate its area A(R). To this end, evaluate
either one of the iterated integrals above with F (x, y) = 1.
3. Sketch a curve given by the equation r = | sin(3φ)| in polar coordinates.
Express the area inside the curve as a double integral in polar coordinates,
then compute the area.
4. Evaluate iterated integrals by converting to polar coordinates:
Z
1
√
Z
(x2 + y 2 )3/2 dydx
−1 0
Z √2 Z
0
1−x2
√
4−y 2
1
p
y
1 + x2 + y 2
dxdy
5. Sketch each of the three described solids, express its volume as a double
integral, then compute the volume:
– inside of x2 + y 2 + z 2 = 9, outside of x2 + y 2 = 1
p
– below z = x2 + y 2 , inside of x2 + y 2 = 2y, above z = 0
– below z = 1 − x2 − y 2 , inside of x2 + y 2 − x = 0, above z = 0
Hint: converting to polar coordinates might be helpful.
6. Sketch the region of integration and evaluate the integrals by converting
to cylindrical coordinates in the first integral and spherical coordinates in
the second one:
Z 3 Z √9−x2 Z 9−x2 −y2
x2 dz dy dx
√
2
0
−3 − 9−x
Z 2 Z √4−x2 Z √4−x2 −y2 p
z 2 x2 + y 2 + z 2 dz dy dx
√
−2
− 4−x2
0
1
7. Use spherical coordinates to find
p the volume of the solid G bounded
16 − x2 − y 2 and below by the cone
above
by
the
hemisphere
z
=
p
2
2
z = x +y .
Exercises for the last week
1*. Evaluate the iterated integral by converting to polar coordinates:
Z 1 Z √y p
x2 + y 2 dxdy .
0
y
√
Answer:
8+2
45 .
2.* Express the area of a given surface as a double integral, and then find the
surface area:
– the portion of the sphere x2 + y 2 + z 2 = 16 between the planes z = 1
and z = 3 Answer: 16π.
– the portion of the surface z = 3x + y 2 that is above the triangular re1
gion with vertices (0, 0), (0, 1) and (1, 1) Answer: 12
(143/2 − 103/2 ).
– the portion of the paraboloid
z = 4 − x2 − y 2 that is above the
√
π
xy-plane Answer: 6 (17 17 − 1).
3.* Use a triple integral to find the volume of the solid within the cylinder
x2 + y 2 = 9 and between the planes z = 1 and x + z = 5. Hint: Try
to sketch the solid, observe that its projection to the xy-plane is the disc
x2 + y 2 ≤ 9, then write the corresponding triple integral. Answer: V =
36π. (See Example 3 in Section 14.5)
4.* Find the volume of the solid G enclosed by the paraboloids z = 5x2 + 5y 2
and z = 6−7x2 −y 2 . Hint: The paraboloids intersect along a curve whose
projection to the xy-plane can be found from 5x2 + 5y 2 = 6 − 7x2 − y 2 .
Therefore projection of G is the region R inside the ellipse 2x2 + y 2 = 1.
3π
Answer: V = √
. ( See Example 4 in Section 14.5)
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5.* Use triple integration in cylindrical coordinates to find thepvolume of the
solid G that is bounded above by the hemisphere z = 25 − x2 − y 2 ,
below by xy-plane, and laterally by the cylinder x2 + y 2 = 9. Answer:
V = 122
3 π. (See Example 1 in Section 14.6)
6.* Draw the region of integration and evaluate the integral by performing a
convenient change of variables:
–
Z Z
x−y
dA
x+y
R
where R is the region enclosed by x − y = 0, x − y = 1, x + y = 1
and x + y = 3. Hint: u = x + y, v = x − y. Answer: 41 ln 3. (See
Example 2 in Section 14.7)
2
–
Z Z
exy dA
R
where R is the region enclosed by the lines y = 21 x and y = x and
the hyperbolas y = x1 and y = x2 . Hint: u = xy , v = xy. Answer:
1 2
2 (e − e) ln 2. (See Example 3 in Section 14.7)
7.* Find the center of gravity of the triangular lamina with vertices (0, 0),
(0,
1) and (1, 0) and density function δ(x, y) = xy. Answer: (x, y) =
2 2
5, 5
. (See Example 2 in Section 14.8)
8.* Find the centroid of the semicircular region
n
o
R = x2 + y 2 ≤ a 2 , y ≥ 0 .
4a
Answer: (x, y) = 0, 3π
. (See Example 3 in Section 14.8)
9.* Find
p the centroid (x, y, z) of the solid G bounded below by the cone z =
x2 + y 2 and above by the sphere x2 + y 2 + z 2 = 16. Hint: You already
computed the volume of G in question 7 of Homework 9. Use symmetry
to find x and y. To find z write thecorresponding
integral in spherical
coordinates. Answer: (x, y, z) = 0, 0, 2(2−3√2) ≈ (0, 0, 2.561). (See
Example 5 in Section 14.8)
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