Volume of a solid of revolution. Hello friends, today I’ll talk about the volume of a solid of revolution. Have a look!!

Want to know more about the volume of a solid? Have a look at:

**The volume of revolutions in polar curves**

**How to apply triple integrals to find out the volume of solids**

**The volume of a solid of revolution**

Now let’s suppose there is a curve . And the plane figure is bounded by the curve , axis and the ordinates and . When this figure is rotated through a complete revolution about the – axis, the volume of revolution of this curve will be

Again suppose there are parametric equations of a curve such as . Then the volume generated when the plane figure bounded by the curve, the axis, the ordinates and rotates about the axis through a complete revolution is

Now I’ll give some examples of that.

**Solved examples of the volume of a solid of revolution**

Disclaimer: None of these examples is 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 area bounded by , the axis and the ordinates , is rotated through a complete revolution about the axis. Show that the volume generated is .”

**Solution**

Now here the equation of the curve is

Also, the area bounded by the given curve, the axis and the ordinates , is rotated through a complete revolution about the axis. And I have to prove that the volume of the solid generated is .

So I’ll start with the value of . And that is

Then I’ll simplify it to get

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

Then I’ll integrate it.

Since is constant, and will also be constants. Thus will be

Now I’ll substitute the limits. So it will be

Then I’ll simplify it to get

So this gives

Also, we all know that

So will be

Therefore the volume of solid generated is .

Hence I have proved it.

Now I’ll give you another example where I’ll use parametric curves.

**Example 2**

According to Stroud and Booth (2013)*, “If , find the volume generated when the plane figure bounded by the curve, the axis and the ordinates at and , rotates about the axis through a complete revolution.”

**Solution**

So in this example, the parametric equations of the curve are

And I have to find out the volume generated when the plane figure bounded by the curve, the axis and the ordinates at and , rotates about the axis through a complete revolution.

Now 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 substitute the values of and to get

Now I’ll simplify it. So that means

Next, I’ll expand .

**Step 2**

Also, we know the formula for as

Thus will be

So will be

Again I know that

and

Therefore I can rewrite as

Next, I’ll simplify it to get

So this means

Then I’ll integrate it.

**Step 3**

Thus I get

Now I’ll substitute the limits to get

Also, I know that

Therefore the value of will be

If I simplify it, I’ll get

Thus the volume 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!!

## Leave a Reply