Area under a parametric curve. Hello friends, today I’ll talk about the area under a parametric curve. Have a look!!

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**How to get the area under a parametric curve**

### The area under a parametric curve

Suppose and are the parametric equations of a curve. Then the area bounded by the curve, the -axis and the ordinates and will be

Now I’ll solve some examples of that.

**Solved examples of the area under a parametric curve **

Note: 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)*, “Determine the area of one arch of the cycloid , i.e. find the area of the plane figure bounded by the curve and the -axis between and .”

**Solution**

Now here the given parametric equations of the cycloid are

And I have to find out the area of the plane figure bounded by the curve and the -axis between and . So I’ll start with the formula

**Step 1**

First of all, I’ll find out the value of in terms of . As I know the value of is

Next, I’ll differentiate with respect to to get

So that gives

Therefore the area bounded by the curve and the -axis between and is

Then I’ll simplify it to get

As per the * standard formulas in trigonometry*,

And that means,

So the area will be

Next, I’ll integrate it.

**Step 2**

And for that, I’ll use the * standard formulas in integration*. Thus it will be

Now I’ll substitute the limits to get

As I know, . So I’ll put these values in the equation of the area and simplify it to get

Therefore the area bounded by the curve and the -axis between 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)*, “The parametric equations of a curve are Find the area under the curve between and .”

**Solution**

Now here the parametric equations of a curve are And I have to find the area under the curve between and . So I’ll start with the formula

**Step 1**

First of all, I’ll find out the value of in terms of . As I know the value of is

Next, I’ll differentiate with respect to to get

So that gives

Therefore the area under the curve between and is

Then I’ll simplify it to get

As per the standard formulas in trigonometry,

And that means,

So the area will be

Next, I’ll integrate it.

**Step 2**

And for that, I’ll use the standard formulas in integration. Thus it will be

Now I’ll substitute the limits to get

Then I’ll simplify it to get

As I know and . So I’ll put these values in the equation of the area. And that gives

Now I’ll simplify it to get

Thus the area under 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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