Surface of revolution of a curve. Hello friends, today I’ll talk about the surface of revolution of a curve. Have a look!!

Want to know more about the surface of revolution? Check out:

**The surface of revolution in polar curves**

**The surface of revolution of a curve**

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 surface of revolution of this curve will be

Again suppose there are parametric equations of a curve such as . Then the area of the surface 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 surface of revolution of a curve**

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 arc of the catenary between and , rotates about the axis. Find the area of the surface so generated.”

**Solution**

Now here the equation of the curve is . Also, I know that the arc of this curve in between and rotates about the axis. And my task is to find out the area of the surface generated, that is, the surface of revolution.

Now the formula for that is

So I’ll start with .

**Step 1**

Since , the value of will be

And that means

Thus I can say that

So will be

Since , I can say that

Therefore will be

Next, I’ll substitute this in the formula for to get

Also, it is already given that . And the limits of integration are . Thus will be

Now I’ll integrate it.

**Step 2**

First of all, I’ll simplify to get

Since , will be

Next, I’ll integrate it to get

Then I’ll substitute the limits. So that gives

Now I’ll simplify it to get

Thus the area of the surface generated is . Hence I can conclude that this is the answer to the given example.

Now I’ll give you another example where I’ll use the curve with parametric equations.

**Example 2**

According to Stroud and Booth (2013)*, “A curve is defined by the parametric equations ; if the arc in between and rotates through a complete revolution about the axis, determine the area of the surface generated.”

**Solution**

Now here the parametric equations of the curve are . Also, the arc of the curve in between and rotates through a complete revolution about the axis. And my task is to find out its surface area of revolution .

Now the formula for that is

So I’ll start with the value of .

**Step 1**

Since , the value of will be

So will be

Again for , the value of will be

Thus will be

Therefore will be

Next, I’ll simplify it to get

Since , I can say that

So will be

Next, I’ll substitute this in the formula for .

**Step 2**

And that gives

Also, it is already given that . And the limits of integration are . Thus will be

Now I’ll simplify it. As I have already mentioned in step 1, . So I can say that

Since , I can say that

Next, I’ll integrate it to get

Then I’ll substitute the limits.

**Step 3**

So that gives

Now I’ll simplify it to get

Also, I already know that . Then I’ll substitute these values to get the value of as

Next, I’ll simplify it. And that gives

Thus the area of the surface 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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