National Institute of Technology Karnataka, Surathkal

Department of Mathematical and Computational Sciences

Engineering Mathematics-I (MA110)

Problem Sheet 8

1. Let D be the region bounded by the paraboloids z = 8 − x2 − y2 and z = x2 + y2 . Write

six different triple iterated integrals for the volume of D. Evaluate one of the integrals.
Z 1 Z 1 Z 1− y
2. Here is the region of integration of the integral dz dy dx
−1 x 2 0

Rewrite the integral as an equivalent iterated integral in the order

(a) dy dz dx (d) dx dz dy
(b) dy dx dz (e) dz dx dy
(c) dx dy dz

3. Find the volume of the region common to the interiors of the cylinders x2 + y2 = 1 and
x2 + z2 = 1.
4. Let D be the region bounded below by the plane z = 0, above by the sphere x2 + y2 + z2 =
4, and on the sides by the cylinder x2 + y2 = 1. Set up the triple integrals in cylindrical
coordinates that give the volume of D using the following orders of integration.

(i) dz dr dθ (ii) dr dz dθ (iii) dθ dz dr

p
5. Let D be the region bounded below by the cone z = x2 + y2 and above by the paraboloid
z = 2 − x2 − y2 . Set up the triple integrals in cylindrical coordinates that give the volume
of D using the following orders of integration.

(i) dz dr dθ (ii) dr dz dθ (iii) dθ dz dr

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 6. Give the limits of integration for evaluating the integral
ZZZ
f (r, θ, z) dz r dr dθ

as an iterated integral over the region that is bounded below by the plane z = 0, on the
side by the cylinder r = cos θ, and on top by the paraboloid z = 3r2 .
7. Convert the integral
Z 1 Z √ 1− y2 Z x  
x2 + y2 dz dx dy
−1 0 0

to an equivalent integral in cylindrical coordinates and evaluate the result.

8. In the following exercises, set up the iterated integral for evaluating
ZZZ
f (r, θ, z) dz r dr dθ
D

over the given region D.

(a) D is the right circular cylinder whose base is the circle r = 2 sin θ in the xy− plane
and whose top lies in the plane z = 4 − y.
(b) D is the right circular cylinder whose base is the circle r = 3 cos θ and whose top lies
in the plane z = 5 − x.
9. In the following exercises, set up the iterated integral for evaluating
ZZZ
f (r, θ, z) dz r dr dθ
D

over the given region D.

(a) D is the solid right cylinder whose base is the region in the xy− plane that lies inside
the cardioid r = 1 + cos θ and outside the circle r = 1 and whose top lies in the plane
z = 4.
(b) D is the prism whose base is the triangle in the xy−plane bounded by the y−axis
and the lines y = x and y = 1 and whose top lies in the plane z = 2 − x.
10. (a) Let D be the region bounded below by the plane z = 0, above by the sphere x2 +
y2 + z2 = 4, and on the sides by the cylinder x2 + y2 = 1. Set up the triple integrals
in spherical coordinates that give the volume of D using the following orders of
integration.

(i) dρ dϕ dθ (ii) dϕ dρ dθ
p
(b) Let D be the region bounded below by the cone z = x2 + y2 and above by the

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 plane z = 1. Set up the triple integrals in spherical coordinates that give the volume
of D using the following orders of integration.

(i) dρ dϕ dθ (ii) dϕ dρ dθ

11. In the following exercises,

(i) find the spherical coordinate limits for the integral that calculates the volume of the
given solid and then
(ii) evaluate the integral.
(a) The solid between the sphere ρ = cos ϕ and the hemisphere ρ = 2, z ≥ 0.
(b) The solid bounded below by the sphere ρ = 2 cos ϕ and above by the cone z =
p
x 2 + y2 .
(c) The solid bounded below by the xy−plane, on the sides by the sphere ρ = 2, and
above by the cone ϕ = π/3.
12. (a) Set up triple integrals for the volume of the sphere ρ = 2 in
(i) spherical,
(ii) cylindrical, and
(iii) rectangular coordinates.
(b) Let D be the smaller cap cut from a solid ball of radius 2 units by a plane 1 unit from
the center of the sphere. Express the volume of D as an iterated triple integral in
(i) spherical,
(ii) cylindrical, and
(iii) rectangular coordinates. Then
(iv) find the volume by evaluating one of the three triple integrals.
13. Find the volumes of the solids in the following exercises.

(a) (b) (c)

14. (a) Sphere and cones : Find the volume of the portion of the solid sphere ρ ≤ a that lies
between the cones ϕ = π/3 and ϕ = 2π/3.
(b) Cylinder and paraboloid : Find the volume of the region bounded below by the

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 plane z = 0, laterally by the cylinder x2 + y2 = 1, and above by the paraboloid
z = x 2 + y2 .
(c) Sphere and cylinder : Find the volume of the region that lies inside the sphere
x2 + y2 + z2 = 2 and outside the cylinder x2 + y2 = 1.
15. (a) Cylinder and planes : Find the volume of the region enclosed by the cylinder x2 +
y2 = 4 and the planes z = 0 and x + y + z = 4.
(b) Region trapped by paraboloids : Find the volume of the region bounded above by
the paraboloid z = 5 − x2 − y2 and below by the paraboloid z = 4x2 + 4y2 .
(c) Paraboloid and cylinder : Find the volume of the region bounded above by the
paraboloid z = 9 − x2 − y2 , below by the xy− plane, and lying outside the cylinder
x2 + y2 = 1.
(d) Sphere and paraboloid : Find the volume of the region bounded above by the
sphere x2 + y2 + z2 = 2 and below by the paraboloid z = x2 + y2 .
16. (a) Find the average value of the function f (r, θ, z) = r over the solid ball bounded by
the sphere r2 + z2 = 1. (This is the sphere x2 + y2 + z2 = 1.)
(b) Find the average value of the function f (ρ, ϕ, θ ) = ρ cos ϕ over the solid solid upper
ball ρ ≤ 1, 0 ≤ ϕ ≤ π/2.
17. Use the transformation
u = 3x + 2y, v = x + 4y

to evaluate the integral ZZ

(3x2 + 14xy + 8y2 ) dx dy
R

for the region R in the first quadrant bounded by the lines y = −(3/2) x + 1, y =
−(3/2) x + 3, y = −(1/4) x, and y = −(1/4) x + 1.
18. Let R be the region in the first quadrant of the xy-plane bounded by the hyperbolas xy =
1, xy = 9 and the lines y = x, y = 4x. Use the transformation x = u/v, y = uv with
u > 0 and v > 0 to rewrite
x √
Z Z r 
+ xy dx dy
y
R

as an integral over an appropriate region G in the uv-plane. Then evaluate the uv-integral
over G.
19. Find the volume of the ellipsoid

x 2 y2 z2
+ 2 + 2 = 1,
a2 b c

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 using a suitable change of variables.
20. Evaluate ZZZ
| xyz| dx dy dz

over the solid ellipsoid

x 2 y2 z2
+ 2 + 2 ≤ 1,
a2 b c
using a suitable substitution.
21. Let D be the region in xyz-space defined by the inequalities

1 ≤ x ≤ 2, 0 ≤ xy ≤ 2, 0 ≤ z ≤ 1.

Evaluate ZZZ
( x2 y + 3xyz) dx dy dz
D

by applying the transformation

u = x, v = xy, w = 3z

and integrating over an appropriate region G in uvw-space.