Apply double integrals to find the area. Hello friends, today I’ll show how to apply double integrals to find the area of a curve. Have a look!!

Want to know more about double or triple integrals? Do check out:

**How to evaluate double integrals?**

**How to evaluate triple integrals?**

### Apply double integrals to find the area of a curve

Now I’ll give you two formulas to find out the area of a curve. And the first one is the polar curve.

Suppose I have a polar curve . And I have to find out the area bounded by the curve and the radius vectors at and . Then the area will be

Next, I’ll give the formula for any curve. And that is

Now I’ll give some examples of that.

**Solved examples of how to apply double integrals to find the area of a 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 my first example.

**Example 1**

According to Stroud and Booth (2013)*, “Form a double integral to represent the area of the plane figure bounded by the polar curve and the radius vectors at and , and evaluate it.”

**Solution**

As I can see, in this example, the plane figure is bounded by the polar curve and the radius vectors at and . First of all, I have to form a double integral to represent the area.

**Step 1**

According to the standard formula, the area of the plane figure in the double integral form will be

Now in this example, and is the equation of the polar curve, i.e., . Since the radius vectors are at and , I can say that .

Therefore the double integral form is

Now I’ll integrate it to get the area of the plane curve.

**Step 2**

First of all, I’ll integrate it with respect to . So it will be

And that means

Next, I’ll substitute the limits. Thus it will be

Now I’ll expand to get

As I know , I can say that

Then I’ll simplify it to get

Next, I’ll integrate it with respect to .

**Step 3**

So this gives

Next, I’ll substitute the limits. Thus it will be

Since and , becomes

Then I’ll simplify it to get

Thus the area of the plane figure 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)*, “Determine the area bounded by the curves and .”

**Solution**

So in this example, I have to find out the area bounded by the curves and . And that means these two curves intersect each other at some points. Also, these points give the limits of the integrations.

Now the formula to determine the area of a curve using double integral is

First of all, I’ll find out the points of intersection of the two curves so that I know the values of .

**Step 1**

So I can say that the two curves are

And for the points of intersection, . Therefore I can say that

Next, I’ll simplify it to get the values of .

Now that means

Thus I can say that either or . So this gives .

Thus the area will be

Next, I’ll do the integration.

**Step 2**

First of all, I’ll integrate with respect to . So it will be

Then I’ll substitute the limits. So it becomes

Now I’ll simplify it to get

Next, I’ll integrate it with respect to . And that means

Then I’ll substitute the limits to get

Now I’ll simplify it to get the value of as

Therefore the area bounded by the curves 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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