Unit-4
Multiple integrals
Question Bank
Question-1: Evaluate
Sol. Let us suppose the integral is I,
I =
Put c = 1 – x in I, we get
I =
Suppose , y = ct
Thendy = c
Now we get,
I =
I =
I =
I =
I =
As we know that by beta function,
Which gives,
Now put the value of c, we get
Question-2: Evaluate-
Where R is the quadrant of the circle .
Sol.
Suppose the region of integration is the first quadrant of the circle OAB.
(
First we will integrate w.r.t. x then w.r.t. y-
The limits for y are 0 and and limits for x are 0 to a.
We get-
Question-3: Evaluate-
By changing into polar coordinates.
Sol.
We have
The limits of x and y are 0 to
Hence the region of integration is in the first quadrant,
The region is covered by the radius strip from r = 0 to r = ∞ and it starts from θ = 0 to θ = π/2
Hence the integral becomes-
Question-4: Evaluate Where V is annulus between the spheres
And ()
Solution: It is convenient to transform the triple integral into spherical polar co-ordinate by putting
, ,
, dxdydz=sindrdd,
and
For the positive octant, r varies from r =b to r =a , varies from
And varies from
I=
= 8
=8
=8
=8
=8 log
= 8 log
I= 8 log I = 4 log
Question-5: Find Volume of the tetrahedron bounded by the co-ordinate planes and the plane
Solution: Volume = ………. (1)
Put ,
From equation (1) we have
V =
=24
=24 (u+v+w=1) By Dirichlet’s theorem.
=24
= == 4
Volume =4
Question-6:
Evaluate
Solution:- Put
Question-7: If , then show that
1.
2.
Sol.
Suppose and
Now taking L.H.S,
Which is
Hence proved.
2.
So that
Question-8: Evaluate , where S is the surface of the sphere in the first octant.
Sol. Here-
Which becomes-
Question-9: Evaluate if V is the region in the first octant bounded by and the plane x = 2 and .
Sol.
x varies from 0 to 2
The volume will be-
Question-10: Evaluate by using Green’s theorem, where C is a triangle formed by
Sol. First we will draw the figure-
Here the vertices of triangle OED are (0,0), (
Now by using Green’s theorem-
Here P = y – sinx, and Q =cosx
So that-
and
Now-
=
Which is the required answer.
Question-11: Prove the following by using Gauss divergence theorem-
1.
2.
Where S is any closed surface having volume V and
Sol. Here we have by Gauss divergence theorem-
Where V is the volume enclose by the surface S.
We know that-
= 3V
2.
Because
Question-12: If and C is the boundary of the triangle with vertices at (0, 0, 0), (1, 0, 0) and (1, 1, 0), then evaluate by using Stoke’s theorem.
Sol. Here we see that z-coordinates of each vertex of the triangle is zero, so that the triangle lies in the xy-plane and
Now,
Curl
Curl
The equation of the line OB is y = x
Now by stoke’s theorem,