Surface Area of Revolution

$Surface Area of Revolution Example 2$

$Surface Area of Revolution Example 1$

Formula for the surface area of revolution

For rotation around the $x$ -axis, $S=\int 2\pi y\ ds$

For rotation around the $y$ -axis, $S=\int 2\pi x\ ds$

where,

$ds=\sqrt{1+\left(\frac{dy}{dx}\right)^2}\ dx$ if $y=f(x)$

$ds=\sqrt{1+\left(\frac{dx}{dy}\right)^2}\ dy$ if $x=h(y)$

When using this formula for the surface area of revolution, first determine whether you need to rotate about the $x$ -axis or about the $y$ -axis. The axis of rotation will dictate which one of the first two pieces of the formula you’ll use. Notice that the variable you see in the formula is the opposite of the axis of rotation. In other words, if you’re rotating about the $x$ -axis, the $y$ variable appears in the integral, and if you’re rotating about the $y$ -axis, the $x$ variable appears in the integral.

Once you identify the axis of rotation (this is often given to you in your problem), and therefore decide which integral formula to use, you can then use either formula for $ds$ . It’s just a matter of which is more convenient, depending on the form in which your function is given to you. Usually, you’ll use the first $ds$ formula if your function is in the form $y=f(x)$ , whereas you’ll use the second $ds$ formula if your function is in the form $x=h(y)$ .

Solving a surface area of revolution problem

Before you get involved with the integral, calculate and simplify $ds$ as much as possible. Remember that, if you’re using the first $ds$ formula because your original function is in the form $y=f(x)$ , then you’ll be calculating $dy/dx$ , which is the derivative of your function $y$ , in terms of $x$ . If you’re using the second $ds$ formula because your original function is in the form $x=h(y)$ , then you’ll be calculating $dx/dy$ , which is the derivative of your function $x$ , in terms of $y$ .

First, calculate $dy/dx$ , or $dx/dy$ , depending on the $ds$ formula you’re using. Second, plug this derivative into your $ds$ formula, and then, according to the formula, square it. Third, you’ll want to create just one fraction underneath the square root, so find a common denominator and combine the two terms underneath the square root to create one fraction. Lastly, now that you have just one fraction inside the square root, take the square root of the numerator and denominator separately. This result is what you want to plug in for $ds$ .

If you used the first formula for $S$ , you’ll now have an integral full of $x$ ‘s, except for the single leftover $y$ variable. If you used the second formula for $S$ , you’ll now have an integral full of $y$ ‘s, except for the single leftover $x$ variable. You now need to substitute your original function for this leftover variable, so that the entire integral is now in terms of one variable.

Finishing the problem is now just a matter of simplifying the integral as much as possible, and then evaluating. Remember that you can pull any constants out in front of the integral, including $2\pi$ , to simplify the integral. Once you integrate, you’ll need to plug in your limits of integration. Keep in mind that these limits of integration are $x$ -values if you are rotating around the $x$ -axis, and $y$ -values if you are rotating around the $y$ -axis.

Surface Area of Revolution

Formula for the surface area of revolution

Solving a surface area of revolution problem

integralCALC

Affiliates

Community

Free eBook