Surface of revolutions in polar curves. Hello friends, today I’ll talk about the surface of revolutions in polar curves. Have a look!!

Want to know more about the polar curves? Check these out…

**How to get the area enclosed by a polar curve?**

**The volume of revolutions in polar curves**

**The surface of revolutions in polar curves**

Now let’s suppose there is a polar curve where is a function of . And the plane figure is bounded by the curve and the radius vectors and . When this figure rotates around the initial line, the surface of revolution of this curve will be

Now I’ll give some examples of that.

**Solved examples on the surface of revolutions in polar curves**

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)*, “Find the area of the surface generated when the arc of the curve between and , rotates about the initial line.”

**Solution**

Now here the equation of the polar curve is

(1)

Also, I have to find out its length in between and .

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

**Step 1**

So, I’ll differentiate equation (1) with respect to to get

Also, from equation (1) I can say that

Therefore the value of will be

Now I’ll simplify it to get

Since , I can say that the value of becomes

So this gives

As I know , the value of will be

Thus will be

Next, I’ll integrate it to get the surface of revolution of the curve.

**Step 2**

Therefore it will be

Now I’ll take out the constant term to get

Also, I already know that and . So in this case, it will be

Then I’ll simplify it to get

Next, I’ll integrate it. And that gives

Now I’ll simplify it to get

Next, I’ll substitute the limits to get the value of as

Also, we all know that and .

And that means the value of will be

Thus the surface of revolution of this curve between and is .

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

Now I’ll give you another example.

**Example 2**

According to Stroud and Booth (2013)*, “Find the area of the surface generated when the arc of the curve between and , rotates about the initial line.”

**Solution**

Now here the equation of the polar curve is

(2)

Also, I have to find out its length in between and .

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

**Step 1**

So, I’ll differentiate equation (2) with respect to to get

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

Therefore the value of will be

So this means

Since , the value of is

Now I’ll simplify it to get

Since , I can say that the value of becomes

Thus will be

Next, I’ll integrate it to get the surface of revolution of the curve.

**Step 2**

Therefore it will be

Since , I can say that

Now I’ll take out the constant term to get

Next, I’ll integrate it. And that gives

Next, I’ll substitute the limits to get the value of as

Also, we all know that and .

And that means the value of will be

Thus the surface of revolution of this curve between and is .

Hence I can conclude that this is the solution 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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