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

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

**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

**Step 1**

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

Next, I’ll differentiate

So that gives

Therefore the area bounded by the curve and the

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,

Therefore the area bounded by the curve and the

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

**Solution**

Now here the parametric equations of a curve are

**Step 1**

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

Next, I’ll differentiate

So that gives

Therefore the area under the curve between

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

Now I’ll simplify it to get

Thus the area under the curve between

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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