Solutions of Practice Questions

Lecture#23

Q1.

Find all possible polar co-ordinates representations of the point P whose rectangular co-ordinates are

Solution

We have

x= and

For polar coordinate

The point P lies in the fourth quadrant

When
When

Polar co-ordinates of P are

Q2.

Find polar co-ordinates of the point whose rectangular co-ordinates are

Solution

As we know

Q3.
Express the equation in polar coordinates.

Solution

We use the formula

Q4.

Convert the point (2, π/3) from polar to Cartesian coordinates.

Solution
Therefore, the point is (1, ) in Cartesian co-ordinates.

Lecture#24

Q1.

Equation of a curve in polar co-ordinates is . Discuss the symmetry of graph of this curve
about initial line.

Solution

Symmetry about initial line

,
As we know so equation of a curve remains same if is replaced by . So the
curve is symmetrical about initial line.

Q2.

Equation of a curve in polar co-ordinates is . Discuss the symmetry of the graph of this
curve about y-axis.

Solution

Symmetry about y-axis

As we know
 So equation of a curve remains same if is replaced by . So the curve is symmetrical
about y-axis.

Q3.

Equation of a curve in polar co-ordinates is . Discuss the symmetry of the graph of this
curve about y-axis.

Solution

Symmetry about y-axis

As we know

So equation of a curve does not remain same if is replaced by . So the curve is not
symmetrical about y-axis.

Lecture#25

Q1.

Use double integrals in polar co-ordinates to find area of the region bounded by the curve .

Solution

Here is a circle of radius and center at

Q2.

In each part, rewrite the double integral as an iterated integral in polar coordinates. (Do not evaluate.)

(a) , where R is the left half of the unit disk.

(b) Where R is the right half of the ring

Solution

(a) The region R is the polar rectangle π/2 ≤ θ ≤ 3π /2, 0 ≤ r ≤ 1. In polar coordinates, the

integrand is . So, we can rewrite the double integral as an iterated integral

 (b) The region R is the polar rectangle − π /2 ≤ θ ≤ π/ 2 , 2 ≤ r ≤ 3. In polar coordinates, the

integrand is . So, we can rewrite the double integral as an iterated integral

Q3.Evaluate the following double integral by changing to polar coordinates.

   

Solution

Q4. Evaluate by converting to polar coordinates.

Solution
Solution of Practice Questions

Lecture#26

Q1. Given the integral Evaluate

the integral with respect to the variable only.

Solution

Q2. Evaluate the double integral by changing to polar co-ordinates.

Solution
Q3. Evaluate the double integral by changing to polar co-ordinates.

Solution
Q4.
Convert the following integral from Cartesian to polar coordinates. (Do not integrate it).

Solution
Lecture#27

Q1. If then find .

Solution

Q2. If the vector valued function , then find the vector that associates
with .

Solution

The vector associated with is

Lecture#28

Q1. Check whether the given limit exists or not? Also justify your answer.

Solution

Since the limit of first component, that is, does not exist. So even though the limit of
remaining two components exist, the limit of vector-valued function does not exist.

Q2. The vector equation for a given curve C is and its derivative is

. Find the tangent vector to C at the point .

Solution

At

For vector equation of tangent line at the point where , we have

Q3.
Let . Find t, such that and are perpendicular to each other.

Solution

Lecture#29
Q#1: Write down the expression for the arc-length of the curve represented by the vector valued
function

where . (Do not evaluate the expression).

Solution:

Here

Q#2: What is the arc-length of the curve ; when ?

Solution:
Q#3: What is the arc-length of the curve when ?

Solution:
Q#4: Given the equations of two curves . Find the intersecting points of these
curves.

Solution:

Q#5: Given the equations of two curves . For which values of x these
curves intersect each other?

Solution:
Q#6: Given the equations of two curves . For which values of x these
curves intersect each other?

Solution:

Lecture#30

Q#1: Determine whether the following differential is exact or not.

Solution:
 3
P=4 x y
2 2
Q=6 x y

∂P
=12 x y2
∂y

∂Q 2
=12 x y
∂x

∂P ∂Q
=
∂ y ∂x

So the given differential is exact.

Q#2: Determine whether the following differential is exact or not.

Solution:

Q#3: Determine whether the following differential is exact? If so, find z.

Solution:
Since the given differential is exact, therefore,
Since is exact differential so

Compare right hand side of both the expressions (1) and (2) we get

Lecture#31

Q#1: Evaluate the line integral along from the point to the
point .

Solution:
Put values of C , x and dx in given line integral

Q#2: Evaluate the line integral where C is the line segment from to .

Solution:

Q#3: Evaluate the line integral along from the point to

the point .

Solution:
Put values of C, y and dy in given line integral

Lecture#32

Q1. Evaluate where C is the curve shown below.

Solution:

So, first we need to parameterize each of the curves

                                              

Now let’s do the line integral over each of these curves.

                            

                                    

Finally, the line integral that we were asked to compute is,

                                         

Answer (-5.732)

Q2. Evaluate the following line integral with respect to arc-lengths

, Where C is the curve .

Solution:
Answer ( )

Q3.

Evaluate the following line integral which is independent of path.

Solution:

Let

As given line integral is independent of path so dz is an exact differential. z will be found as given
below
 As there is no dissimilar term at the right side of equations (1) and (2), so

So for , we have

Put this value in given integral, with limits to

Answer (32)

Q4.

Evaluate the following line integral which is independent of path.

Solution:

Let
 Since given line integral is independent of path so dz is an exact differential. z will be found as
given below.

As there is no dissimilar term at the right side of equations (1) and (2) so

So for , we have

Put this value in given integral, with limits

to

Answer (1)
Lecture#33

Q1. Use Green’s Theorem to evaluate , where C is the triangle with vertices
with positive orientation.

 Solution:

So, the curve does satisfy the conditions of Green’s Theorem and we can see that the following
inequalities will define the region enclosed.

                                              

We can identify P and Q from the line integral.  Here they are.

                                                    

 
So, using Green’s Theorem the line integral becomes,

                                         

Answer ( )

Q2.

Use Green’s Theorem to evaluate

where C is a square with vertices at

, , and

Solution:

Here
Limits for x is from 0 to 2 and for y is .

Thus by Green’s Theorem

Answer (20)

Q3. Use Green’s Theorem to evaluate   where C is the positively oriented circle of radius
2 centered at the origin.

Solution:
Let’s first identify P and Q from the line integral.

Now, using Green’s theorem on the line integral gives,

                                              

Where D is a disk of radius 2 centered at the origin. 

Since D is a disk it seems like the best way to do this integral is to use polar coordinates.  Here is the
evaluation of the integral.

                                             

Answer ( )

Lecture#34

Q1. Find , if .

Solution:
 Answer ( )

Q2. Find , if

Solution:
Answer

( )

Q3. If then find .

Solution:

Answer ( )

Q4:

Solution:
We are given that
Answer ( )

Question 1

Evaluate by parts.

Ans

Question 2

Evaluate by parts.

Ans

Question 3

Evaluate
Lecture#35

1. Use Wallis cosine formula to evaluate .

Solution

2. Use Wallis sine formula to evaluate .

Solution:
Lecture#36

1. Evaluate the line integral where and c is given by the

curve .

Solution:
We are given that and .
This implies
.
Also,

which is the required result.

 2. Evaluate the line integral where and c is the
curve given by .

Solution:

We are given that and .

.
Also,

Lecture#37

Consider the vector field over the surface S which is the portion of the sphere

bounded by the planes (as shown in the figure below).

If is a unit upper normal to S. express in spherica coordinates.

Note: Do not evaluate the integral.

Solution:

Put values of and in given integral

 To convert it to spherical polar coordinates, we have

Limits are:

Put these values

Practice Questions

Lecture#38

Q1. Determine whether the following vector field is conservative or not.

 
Q2. Let is the surface bounded by the plane . If

, use Divergence Theorem to find , where denote the

unit outer normal to S.

Q3.
Use Stokes’ Theorem to evaluate the integral

If and surface S is the portion of paraboloid above the plane

. Assume that S is oriented upward.

Lecture#39

Q1. Write amplitude, period and frequency of the following:

c)

 
Q2. Define the periodic function whose graph is shown below.

Q3. Sketch the graph of the following periodic functions showing all relevant values:

Q4. Evaluate

Lecture#40

Q1. Determine the Fourier co-efficient of the following function.

Q2. Determine the Fourier co-efficient of the following function.

Q3. Determine the Fourier co-efficient of the following function.

Q4. Determine the Fourier co-efficient , and of the following function

Practice Questions Lecture#41-43

Q1. Check whether the given function is even, odd or neither.

Solution

Q2. Check whether the given function is even, odd or neither.

.
Solution
Q3. Consider a periodic function defined by

Determine Fourier Co-efficient , and

Solution

If f(x) is an odd function defined over the interval – π < x < π, then the Fourier series for f(x)

contains sine terms only. So are zero.

Q4. Determine the Fourier coefficient for a periodic function of period 4 defined by

Solution
 Here the given function is of period 4, so T = 4 and

Q5. Determine the Fourier co-efficient of the periodic function

Solution
Q6. Determine the Fourier co-efficient of the following periodic function.

Solution
Practice Questions with Solution

Lecture#44 &45

Q1. Find Laplace transform of the functions if

Solution:

Solution:
Solution:

Solution:

Solution:

Q2. Show that Laplace transform of the function

is where s is a constant for the integration and s > 0.

Solution:
Q3. Find Laplace transform of the function if

Solution:
Q4. Determined:

Solution:

Solution:
 Q5. Find Laplace transform of the function if

Solution: