Line integral of a scalar field. Hello friends, today it’s about the line integral of a scalar field.

Line integral of a scalar field_compressed

**The line integral of a scalar field**

**Solved examples of the line integral of a scalar field**

Disclaimer: None of these examples are mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

So here is the first example.

** Example 1**

According to Stroud and Booth (2011),* “If , evaluate

**Solution**

Here the given scalar field is

Now my first step will be to write

**Step 1**

So, I’ll substitute

Then it will be

Now I’ll simplify it. So it will be

And this gives the value of

Next I’ll get the values of

Since

As I know

Also, since

Now I have to get the value of

Thus in terms of

So this means

Now my next step is to find out the limits of integration.

**Step 2**

Ok, so I have to integrate the scalar field

So, I already know that the

Thus it becomes

Now I’ll check if

If I put

So this gives the

Similarly, I put

So here also I get the

In the same way, now I’ll get the value of

which gives

So I have two limits of integration – one is

Thus my next step will be to evaluate

**Step 3**

So I can rewrite

Now I’ll integrate this vector in the same way as the integration of a vector field. Also, I’ll follow the same rules as the rules for integration.

Hence it will be

Next, I’ll substitute these limits to get

And this gives

If I simplify it, I’ll get

Hence I can conclude that this is the answer to the given example.

Now I’ll give another example.

** Example 2**

According to Stroud and Booth (2011),* “If

**Solution**

Here the given scalar field is

Now my first step will be to write

**Step 1**

So, I’ll substitute

Then it will be

Now I’ll simplify it a bit to get

Now I’ll get the values of

Since

As I know

Also, since

Next, I’ll get the value of

Thus in terms of

So this means

Now my next step is to find out the limits of integration.

**Step 2**

Ok, so I have to integrate the scalar field

So, I already know that the

Now I’ll check if

If I put

So this gives the

Similarly, I put

So here also I get the

In the same way, now I’ll get the value of

which gives

So I have two limits of integration – one is

Thus my next step will be to evaluate

**Step 3**

So I can rewrite

Now I’ll integrate this vector in the same way as in example 1.

Hence it will be

Next, I’ll substitute these limits to get

And this gives

If I simplify it, I’ll get

Hence I can conclude that this is the answer to the given example.

Now I’ll give my last example.

** Example 3**

According to Stroud and Booth (2011),* “Evaluate to one decimal place the integral

**Solution**

Here the given scalar field is

Now my first step will be to write

**Step 1**

So, I’ll substitute

Thus it will be

And this means

Now I’ll get the values of

Since

As I know

Also, since

Next, I’ll get the value of

Thus in terms of

Now my next step is to find out the limits of integration.

**Step 2**

Ok, so I have to integrate the scalar field

So, I already know that the

Now I’ll check if

If I put

So this gives the

Similarly, I put

So here also I get the

In the same way, now I’ll get the value of

which gives

So I have two limits of integration – one is

Thus my next step will be to evaluate

**Step 3**

So I can rewrite

where

and

Now I’ll integrate these vectors in the same way as in examples 1 and 2.

Hence it will be

Next, I’ll substitute these limits to get

Then I’ll simplify it to get

(1)

Now I’ll get the value of

###### Step 4

Here also I’ll do it in the same way as

Next, I’ll substitute these limits to get

Then I’ll simplify it to get

(2)

Finally I’ll add equations (1) and (2) to get the value of

And this means

Hence I can conclude that this is the answer to the given example.

Dear friends, this is the end of today’s post on how to get the line integral of a scalar field. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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