WTW258

CALCULUS

Lecture Units 2.3 and 2.4

1. Let E be the tetrahedron in the first octant bounded by the coordinate planes and
the plane x + 2y + 3z = 6.

(a) List 6 different iterated integrals that represent the volume of E.

(b) Find the volume of E. [6]
(c) Give an iterated integral that may be used to find the mass of E if the density
at any point is equal to the distance from the point to the xz-plane.

2. Give an iterated integral that may be used to find the volume of the solid region
bounded by the parabolic cylinder y = x2 and the planes y + z = 4 and z = 0 with
order of integration
(a) dz dx dy (b) dz dy dx (c) dx dy dz

3. Give an iterated integral in Cartesian coordinates that may be used to find the
volume of the solid region

(a) bounded by the elliptic cylinder x2 +9y 2 = 9 and the planes z = 0 and z = x+3.
(b) bounded by the paraboloid z = 4x2 +y 2 and the parabolic cylinder z = 4−3y 2 .
(c) bounded above by the sphere x2 + y 2 + z 2 = 2a2 and below by the paraboloid
az = x2 + y 2 (a > 0).

4. Give an iterated integral in cylindrical coordinates that may be used to find the
volume of the given solid region:

(a) The region inside the sphere x2 +y 2 +z 2 = 9 and inside the cylinder x2 +y 2 = 4.
(b) The region in the first octant that is inside the sphere x2 + y 2 + z 2 = 16 and
inside the cylinder x2 + y 2 = 4x.
(c) The region bounded by z = x2 + y 2 − 4 and z = −x2 − y 2 .
p
(d) The region between z = x2 + y 2 and z = 0 and inside x2 + (y − 1)2 = 1.
(e) The region bounded by y = 4 − x2 − z 2 and the xz-plane.

5. Give an iterated integral in spherical coordinates that may be used to find the
volume of the given solid region:
p
(a) The region above the cone z = 2 x2 + y 2 and between the spheres
x2 + y 2 + z 2 = 1 and x2 + y 2 + z 2 = 4.

1
 2 2 2
(b) The region above the
pxy-plane that is inside the sphere x + y + z = 8 and
2
below the cone z = 3x + 3y .2

(c) The region inside the sphere x2 + y 2 + z 2 = 4z and above the plane z = 1.

6. Give an iterated integral in (1) cylindrical coordinates and (2) spherical coordinates
that may be used to find the volume of the given solid region:
p p
(a) The region bounded by z = x2 + y 2 and z = 4 − x2 − y 2 .
p
(b) The region above z = − 4 − x2 − y 2 , below the xy-plane and outside
x2 + y 2 = 3.
p
(c) The region bounded by the cone z = x2 + y 2 , the cylinder x2 + y 2 = 4 and
the xy-plane.
p
(d) The region inside x2 + y 2 + z 2 = 10z and above z = 2 x2 + y 2 .
(The curve of intersection is x2 + y 2 = 16, z = 8.)

7. Give an iterated integral that may be used to find the mass of the given solid region:

(a) The region that is bounded by the paraboloid z = 4 − x2 − y 2 and the xy-plane
if the density at any point is directly proportional to the distance from the
point to the xy-plane.
(b) The region in the first octant bounded by the paraboloid z = 4 − 9x2 − y 2 and
the planes y = 4x, z = 0 and y = 0 if the density at any point P is equal to
the square of the distance from P to the origin.
(c) The region cut out of the sphere x2 + y 2 + z 2 = 4 by the cylinder x2 + y 2 = 2y
if the density at any point P is equal to the distance from P to the xy plane.
p
(d) The region that is bounded by the cone z = x2 + y 2 and the plane z = 4 if
the density at any point is directly proportional to the distance from the point
to the z-axis.
(e) The region that is outside the sphere x2 + y 2 + z 2 = 1 but inside the sphere
x2 + y 2 + z 2 = 4 if the density at any point is equal to the distance from the
point to the center of the spheres.

8. Rewrite the integral as an iterated integral in the indicated coordinate system:

Z Z √ 2 Z √ 4−y 2 8−x2 −y 2
(a) √ z 2 dz dx dy in cylindrical and spherical coordinates.
0 0 x2 +y 2
Z 1 Z √
1−x2 Z √1−x2 −y2 p
(b) x2 + y 2 + z 2 dz dy dx in cylindrical and spherical co-
−1 0 0
ordinates.

2
 √
Z 3 Z 9−x2 Z x2 +y 2
(c) √
(x2 + y 2 ) dz dy dx in cylindrical and spherical coordinates.
−3 − 9−x2 0
√
Z 2Z 2x−x2 Z x2 +y 2 p
(d) √
x2 + y 2 dz dy dx in cylindrical coordinates.
0 − 2x−x2 0
Z √
2 Z √4−y2 Z √4−x2 −y2 p
(e) x2 + y 2 + z 2 dz dx dy in spherical coordinates.
0 y 0

9. Find the volume enclosed by the spheres x2 + y 2 + z 2 = 9 and x2 + y 2 + (z − 2)2 = 4.

[ 63π
8
]

Other Answers for Units 2.3 and 2.4

Z 6 Z 1 (6−x) Z 1 (6−x−2y) Z 3 Z 6−2y Z 1 (6−x−2y)
2 3 3
1 (a) 1 dz dy dx = 1 dz dx dy
0 0 0 0 0 0
Z 3 Z 1 (6−2y) Z 6−2y−3z Z 2 Z 1 (6−3z) Z 6−2y−3z
3 2
= 1 dx dz dy = 1 dx dy dz
0 0 0 0 0 0
Z 6 Z 1 (6−x) Z 1 (6−x−3z) Z 2 Z 6−3z Z 1 (6−x−3z)
3 2 2
= 1 dy dz dx = 1 dy dx dz
0 0 0 0 0 0
1 1
Z 6 Z
2
(6−x) Z
3
(6−x−2y)
1 (c) y dz dy dx
0 0 0
√ √
Z 4 Z y Z 4−y Z 2 Z 4 Z 4−y Z 4 Z 4−z Z y
2 (a) √
1 dz dx dy ; (b) 1 dz dy dx ; (c) √
1 dx dy dz
0 − y 0 −2 x2 0 0 0 − y

Z 1 Z 3√1−y2 Z x+3 Z 1 Z √
1−x2 Z 4−3y 2
3 (a) √ 1 dz dx dy ; (b) √
1 dz dy dx ;
−1 −3 1−y 2 0 −1 − 1−x2 4x2 +y 2
Z a Z √ a2 −x2 Z √2a2 −x2 −y2
(c) √
1 dz dy dx
1
−a − a2 −x2 a
(x2 +y 2 )
√ π
√
Z 2π Z 2 Z 9−r2 Z
2
Z 4 cos θ Z 16−r2
4 (a) √
1.r dzdrdθ ; (b) 1.r dzdrdθ ;
0 0 √ − 9−r2 0 0 0
Z 2π Z 2 Z −r2 Z π Z 2 sin θ Z r
(c) 1.r dzdrdθ ; (d) 1.r dzdrdθ ;
0 0 r2 −4 0 0 0
Z 2π Z 2 Z 4−r2
(e) 1.r dydrdθ
0 0 0
√
Z 2π Z tan−1 ( 21 ) Z 2 Z 2π Z π
2
Z 8
2
5 (a) 1.ρ sin φ dρdφdθ ; (b) 1.ρ2 sin φ dρdφdθ ;
π
0 0 1 0 6
0
π
Z 2π Z
3
Z 4 cos φ
(c) 1.ρ2 sin φ dρdφdθ
0 0 sec φ

3
 √ √ π
Z 2π Z 2 Z 4−r2 Z 2π Z
4
Z 2
6 (a) 1.r dzdrdθ ; 1.ρ2 sin φ dρdφdθ
0 0 r 0 0 0
2π
Z 2π Z 2 Z 0 Z 2π Z
3
Z 2
(b) √ √
1.r dzdrdθ ; √ 1.ρ2 sin φ dρdφdθ
π 3
0 3 − 4−r2 0 2 sin φ
π 2
Z 2π Z 2 Z r Z 2π Z
2
Z
sin φ
(c) 1.r dzdrdθ ; 1.ρ2 sin φ dρdφdθ
π
0 0 0 0 0
√ 4
Z 2π Z 4 Z 5+ 25−r2 Z 2π Z tan−1 ( 12 ) Z 10 cos φ
(d) 1.r dzdrdθ ; 1.ρ2 sin φ dρdφdθ
0 0 2r 0 0 0

Z 2π Z 2 Z 4−r2 Z 8 Z 1 √4−y2 Z 4−9x2 −y 2

5 3
7 (a) kz.r dzdrdθ ; (b) (x2 + y 2 + z 2 ) dzdxdy
y
0 0 0 0 0
√ 4
Z π Z 2 sin θ Z 4−r2 Z 2π Z 4 Z 4
(c) √
|z|.r dzdrdθ ; (d) kr2 dzdrdθ ;
Z0 2π Z0 − 4−r2 0 0 r
π Z 2
(e) ρ3 sin φ dρdφdθ
0 0 1
π
√ π π
√
Z
2
Z 2 Z 8−r2 Z
2
Z
4
Z 8
8 (a) z 2 r dzdrdθ =
ρ4 cos2 φ sin φ dρdφdθ
Z0 π Z 01 Z r√1−r2 √ 0 0
Z π0 Z π Z 1
2
(b) r2 + z 2 r dzdrdθ = ρ3 sin φ dρdφdθ
Z 02π Z0 3 Z0 r2 Z 2π Z π 0
Z 03 0
2 sin φ
(c) r3 dzdrdθ = ρ4 sin3 φ dρdφdθ
cos φ
0 0 0 0 tan−1 ( 13 )
sin2 φ
π π π
Z
2
Z 2 cos θ Z r2 Z
4
Z
2
2Z
2
(d) r dzdrdθ ; (e) ρ3 sin φ dρdφdθ
− π2 0 0 0 0 0

Z 2π Z cos−1 ( 34 ) Z 3 Z 2π Z π
2
Z 4 cos φ
2
9 ρ sin φ dρdφdθ + ρ2 sin φ dρdφdθ
0 0 0 0 cos−1 ( 34 ) 0