Area enclosed by a polar curve. Today I’ll show how to get the area enclosed by a polar curve. Have a look!!

**The area enclosed by a polar curve**

Let’s suppose I have polar curve where is the function of . So the area enclosed by the curve and the radius vectors at and will be

Now I’ll give you some examples of that.

**Solved examples of the area enclosed by 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 are the examples.

**Example 1**

According to Stroud and Booth (2013)*, “…find the area bounded by the curve and the radius vector at and .”

**Solution**

Now here the equation of the curve is . And I have to get the area bounded by this curve and the radius vector at and .

As per the formula,

So in this example, will be . Thus is

Also, and .

(1)

Since it’s not possible to integrate straightaway, I’ll write it in terms of cosine of angles.

**Step 1**

So it will be

Also, from the * standard formulas in trigonometry*, I know that

Therefore in this example, it will be

Next, I’ll simplify it to get

Again I know that

Thus the term will be .

So becomes

Now I’ll simplify it to get

Next, I’ll substitute this value of in equation (1) to get

Then I can integrate it easily.

**Step 2**

So that gives

Now I’ll simplify it to get

Next, I’ll substitute the limits with the upper limit as and lower limit as . So becomes

Also, I already know that .

Thus the area will be

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

Now I’ll give you another example.

**Example 2**

According to Stroud and Booth (2013)*, “Determine the area bounded by the curve and the radius vectors at and .”

**Solution**

Now here the given equation of the curve is . And I have to get the area bounded by this curve and the radius vectors at and . As in example 1, here also I’ll get the value of first.

**Step 1**

Since , the value of will be

Now I’ll simplify it to get

Again I can rewrite it as

Also, we all know that , and . Therefore will be

Next, I’ll simplify it. And that gives

Now I’ll find out the area bounded by this curve and the radius vectors at and .

**Step 2**

So it will be

And this means

Next, I’ll integrate it to get

Then I’ll simplify it. And that means

Now I’ll substitute the limits with the upper limit as and the lower limit as . So becomes

Next, I’ll simplify it to get

Also, I already know that and .

Thus the area will be

Then I’ll simplify it to get

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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