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

### Length of any curve

Let’s suppose is the equation of any curve. Then the length of the curve between and is

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

#### Solved examples on the length of any 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)*, “Find the length of the arc of the curve , between and .”

**Solution**

Now here the equation of the curve is

(1)

First of all, I’ll get the value of .

**Step 1**

So, from equation (1), I can say that the value of is

Next, I’ll differentiate with respect to to get

Then I’ll simplify it to get

So this means

Therefore the value of is

If I simplify it, I’ll get

Next, I’ll get the value of .

**Step 2**

At first, I’ll get the value of .

Thus it will be

Now I’ll simplify it to get

So this gives

Therefore the value of is

And that means

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

###### Step 3

So it will be

And that means

Next, I’ll integrate it to get

See also: **Standard formulas in integration**

Then I’ll substitute the limits to get

Now I’ll simplify it. And that gives

Thus the length of the arc 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)*, “Find the length of the curve between and .”

**Solution**

Now here the equation of the curve is

(2)

First of all, I’ll get the value of .

**Step 1**

So I’ll differentiate equation (2) throughout with respect to . And that gives

Next, I’ll simplify it to get

So this means

And that gives the value of as

Now I’ll get the value of . So that will be

Next, I’ll get the value of .

**Step 2**

At first, I’ll get the value of .

Thus it will be

Now I’ll simplify it to get

So this gives

Therefore the value of is

And that means

Now I’ll get the the length of the curve between and .

###### Step 3

So it will be

And that means

Next, I’ll integrate it to get

Then I’ll substitute the limits to get

Now I’ll simplify it and that gives

Thus the length of the curve 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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