James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

8. Show that
y`
0
arctan x 2 arctan x
x
dx − 2 ln
by first expressing the integral as an iterated integral.
9. (a) Show that when Laplace’s equation
− 2 u
−x 1 −2 u
2 −y 1 −2 u
2 −z − 0 2
is written in cylindrical coordinates, it becomes
− 2 u
−r 2 1 1 r
−u
−r 1 1 −2 u
r 2 − 1 −2 u
2 −z − 0 2
(b) Show that when Laplace’s equation is written in spherical coordinates, it becomes
− 2 u
− 2 1 2
−u
− 1 cot
2 −u
− 1 1 2 −2 u
− 2 1 1
2 sin 2
− 2 u
− 2 − 0
10. (a) A lamina has constant density and takes the shape of a disk with center the origin and
radius R. Use Newton’s Law of Gravitation (see Section 13.4) to show that the magnitude
of the force of attraction that the lamina exerts on a body with mass m located at
the point s0, 0, dd on the positive z-axis is
F − 2GmdS 1 d 2 1
D
sR 2 1 d 2
[Hint: Divide the disk as in Figure 15.3.4 and first compute the vertical component of
the force exerted by the polar subrectangle R ij.]
(b) Show that the magnitude of the force of attraction of a lamina with density that occupies
an entire plane on an object with mass m located at a distance d from the plane is
F − 2Gm
Notice that this expression does not depend on d.
11. If f is continuous, show that
12. Evaluate lim n 22
n l `
o n
o n 2
i−1 j−1
13. The plane
y x
0 yy 0 yz 0
f std dt dz dy − 1 2 y x
sx 2 td 2 f std dt
1
sn 2 1 ni 1 j .
x a 1 y b 1 z − 1 a . 0, b . 0, c . 0
c
0
cuts the solid ellipsoid
x 2
a 2 1 y2
b 2 1 z 2
c 2 < 1
into two pieces. Find the volume of the smaller piece.
1066
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