Length of a parametric curve. Hello friends, today it’s all about the length of a parametric curve. Have a look!!

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**How to get the length of a curve**

**Length of a parametric curve**

Suppose and are the parametric equations of a curve. Then the length of the curve between and is

Now I’ll give some examples on the length of a parametric curve.

**Solved examples of the length of a parametric 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 the first example.

**Example 1**

According to Stroud and Booth (2013)*, “If the parametric equations of a curve are , show that the length of arc between points corresponding to and is .”

**Solution**

Now here the parametric equations of the curve are

(1)

(2)

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

**Step 1**

So I’ll differentiate equation (1) with respect to . And that gives

Then I’ll simplify it to get

Thus will be

(3)

Next, I’ll differentiate equation (2) with respect to . And that gives

Then I’ll simplify it to get

Therefore will be

(4)

Then I’ll add equations (3) and (4) to get the value of .

**Step 2**

Thus it will be

And this is because .

Next, I’ll simplify it to get the value of . And that means

Since , I can say that .

Therefore will be

(5)

Next, I’ll get the length of the arc between points corresponding to and .

**Step 3**

Now as per the formula for the length of the curve , it will be

Then I’ll substitute the value of from equation (5) to the value of as

And that means

Now I’ll integrate it with respect to to get

Next, I’ll substitute the limits to get

Since , the value of will be

So I have proved that the length of the arc between points corresponding to and is .

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 (2013)*, “A curve is defined by the parametric equations :

(a) Determine the length of the curve between and .”

**Solution**

Now here the parametric equations of the curve are

(6)

(7)

And I have to find out the length of the curve between and .

First of all, I’ll get the value of .

**Step 1**

Now I’ll differentiate equation (6) with respect to to get

Thus will be

(8)

Next, I’ll differentiate equation (7) with respect to to get

(9)

Now I’ll add equations (8) and (9) to get

Then I’ll simplify it to get

Next, I’ll get the length of the curve between and .

**Step 2**

Thus it will be

So this means

Now . So it will be

Then I’ll integrate it to get

Next, I’ll substitute the limits to get

And that means

As I already know and , the value of will be

Thus the length of the curve between and is .

Hence I can conclude that this is the answer 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!!

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