Scribd
Upload
91 views2 pages

Practice Problem Set 6

This document contains a practice problem set for a Real Analysis-II course, focusing on double and triple integrals, volume calculations, and the application of Green's theorem. It includes various problems involving functions defined over specific regions, volume of solids bounded by surfaces, and integrals in both cylindrical and spherical coordinates. The problems are designed to enhance understanding of integration techniques in multivariable calculus.

Uploaded by

Mayank Choudhary
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
91 views2 pages

Practice Problem Set 6

This document contains a practice problem set for a Real Analysis-II course, focusing on double and triple integrals, volume calculations, and the application of Green's theorem. It includes various problems involving functions defined over specific regions, volume of solids bounded by surfaces, and integrals in both cylindrical and spherical coordinates. The problems are designed to enhance understanding of integration techniques in multivariable calculus.

Uploaded by

Mayank Choudhary
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 2



MM 451 Real Analysis-II


Practice Problem Set - 6

Instructor: Dharmendra Kumar UOH, Hyderabad

1. Make a sketch of the ordinate set of f over the rectangle Q and compute the volume of this
ordinate set by double integration, when f as follows:

1 − x − y if x + y ≤ 1,
(a) f : Q = [0, 1] × [0, 1] −→ R such that f (x, y) =
0 otherwise.

x + y if x2 ≤ y ≤ 2x2 ,
(b) f : Q = [0, 1] × [0, 1] −→ R such that f (x, y) =
0 otherwise.
 2
x + y 2 if x2 + y 2 ≤ 1,
(c) f : Q = [−1, 1] × [−1, 1] −→ R such that f (x, y) =
0 otherwise.

(x + y)−2 if x ≤ y ≤ 2x,

(d) f : Q = [1, 2] × [1, 4] −→ R such that f (x, y) =
0 otherwise.
2. Let D be the solid bounded by the cylinder x2 + y 2 = 1 and the planes y + z = 1 and z = 0.
Find the volume of D.

3. Let D be the solid bounded by the surfaces y = x2 , y = 3x, z = 0 and z = x2 + y 2 . Find the
volume of D.
n o RR
4. Let R = (x, y) ∈ R2 : 9x2 + 4y 2 ≤ 1 . Find the R sin (9x2 + 4y 2 ) dxdy.

5. Find the volume of the solid bounded by the surfaces z = 3(x2 + y 2 ) and z = 4 − (x2 + y 2 ).
H
6. Use Green’s theorem to evaluate the line integral C y 2 dx + x dy, when:

(a) C is the square with vertices (0, 0), (2, 0), (2, 2) and (0, 2).

(b) C is the square with vertices (±1, ±1).

(c) C is the square with vertices (±2, 0) and (0, ±2).

(d) C is the circle of radius 2 and center at the origin.

(e) C has the vector equation α(t) = 2 cost, 2 sin3 t , 0 ≤ t ≤ 2π.




7. Using the change of variable formula in polar coordinate (in a Double Integral), find the volume
of the sphere of radius a.
x2 y2
8. Find the area bounded by the ellipse C := a2
+ b2
= 1.

9. Evaluate the following triple integral and make a sketch of the region of integration in each
case:

(a)
ZZZ
x y 2 z 3 dx dy dz
S
where S is the solid bounded by the surface z = xy and the planes y = x, x = 1 and z = 0.

(b)
ZZZ
x dx dy dz
S
where S is the solid bounded by the surface z = x2 + y 2 and the planes x = 0, y = 0 and z = 3.

(c)
ZZZ
1
−3
dx dy dz
S (1 + x + y + z)
where S is the solid bounded by the coordinate planes x = 0, y = 0, z = 0 and the plane
x + y + z = 1.

10. Evaluate the following integral by transforming to cylinderical coordinates:

(a)
ZZZ
(x2 + y 2 ) dx dy dz
S
where S is the solid bounded by the surface 2z = x2 + y 2 and the plane z = 2.

(b)
ZZZ
dx dy dz
S
where S is the solid bounded by the three coordinate planes, surface z = x2 + y 2 and the plane
x + y = 1.

(c)
ZZZ
(z 2 x2 + z 2 y 2 )dx dy dz
S
where S is the solid bounded by x + y 2 ≤ 1, −1 ≤ z ≤ 1.
2

11. Evaluate the following integral by transforming to spherical coordinates:

(a)
ZZZ
dx dy dz
S
where S is sphere of radius a and center at the origin.

(b)
ZZZ
dx dy dz
S
where S is region bounded by two concentric spheres of radius a and b, 0 < a < b center at

the origin .

All the best !

You might also like