Length of a polar curve. Hello friends, today I’ll show how to find out the length of a polar curve. Have a look!!

Want to know more about the polar curves? Check it out…

**How to get the area enclosed by a polar curve?**

**Length of a polar curve**

Now let’s suppose there is a polar curve where is a function of . And the length of this curve in between and will be

Now I’ll give some examples of that.

**Solved examples of the length of a polar curve**

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 my first example.

**Example 1**

According to Stroud and Booth (2013)*, “Find the length of the polar curve between and .”

**Solution**

Now here the equation of the polar curve is

(1)

Also, I have to find out its length in between and .

First of all, I’ll find out the value of .

**Step 1**

So, I’ll differentiate equation (1) with respect to to get

Next, I’ll simplify it to get

Thus the value of will be

Also, from equation (1) I can say that

Therefore the value of will be

Now I’ll simplify it to get

Thus will be

So this means

Next, I’ll integrate it to get the length of the curve.

**Step 2**

Therefore it will be

And that gives

Now I’ll simplify it to get

Next, I’ll substitute the limits to get the value of as

Also, we all know that and .

And that means the value of will be

Thus the length of this curve between and is .

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

Now I’ll give you another example.

**Example 2**

According to Stroud and Booth (2013)*, “Find the length of the arc of the curve between and .”

**Solution**

Now here the equation of the polar curve is

(2)

Also, I have to find out its length in between and .

First of all, I’ll find out the value of .

**Step 1**

So, I’ll differentiate equation (2) with respect to to get

Next, I’ll simplify it to get

Thus the value of will be

Also, from equation (2) I can say that

Therefore the value of will be

Now I’ll simplify it to get

Thus will be

So this means

Next, I’ll integrate it to get the length of the curve.

**Step 2**

Therefore it will be

Now I can rewrite it as

So this gives

Next, I’ll integrate it. And that gives

Now I’ll simplify it to get

Next, I’ll substitute the limits. And that gives

Then the value of will be

Also, we all know that .

And that means the value of will be

If I simplify it, I’ll get the value of as

Thus the length of this curve between and is .

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

Dear friends, this is the end of today’s post. 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!!

## Leave a Reply