Volume of revolutions in polar curves. So today I’ll talk about the volume 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?**

Volume of revolutions in polar curves_compressed

**The volume 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 volume of revolution of this curve in between will be

Now I’ll give some examples of that.

**Solved examples of the volume 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)*, “The plane figure bounded by the curve and the radius vectors at and , rotates about the initial line through a complete revolution. Determine the volume of the solid generated.”

**Solution**

Now here the equation of the polar curve is . Also, the plane figure bounded by the curve and the radius vectors at and rotates about the initial line through a complete revolution. And I have to find out the volume of the solid thus generated.

So I’ll start with the value of .

**Step 1**

Since , the value of will be

Next, I’ll use the formula for the volume of the solid generated. Thus it will be

Then I’ll integrate it.

**Step 2**

First of all, I’ll take out the constants. So it will be

But I can also rewrite it as

And that means

Also I can say that

Now I’ll substitute the limits. So that gives

Also I know that and . So that gives

Next, I’ll simplify it to get

So the volume of the solid generated 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 volume generated when the plane figure enclosed by the curve between and , rotates around the initial line.”

**Solution**

Now here the equation of the polar curve is . Also, the plane figure bounded by the curve and the radius vectors at and rotates about the initial line through a complete revolution. And I have to find out the volume of the solid thus generated.

So I’ll start with the value of .

**Step 1**

Since , the value of will be

Next, I’ll use the formula for the volume of the solid generated. Thus it will be

Then I’ll simplify it. So it will be

As I know , I can say that . Thus will be

Now I’ll integrate it.

**Step 2**

First of all, I’ll take out the constants. So it will be

And that means

Also I can rewrite it as

Now I’ll substitute the limits. So that gives

Also I know that and . So that gives

Next, I’ll simplify it to get

So the volume of the solid generated 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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