How to Solve the Surface Area of a Solid of Revolution?
We can determine the surface area generated by a solid of revolution using definite integrals. In this post, let's explore how it expands the approximate solution.
A unique application of integration is finding the surface area of a solid, specifically a solid of revolution. How are we to accomplish this using integration?
Approximation
Let's first find the surface area by approximation. We imagine its surface area as a series of bands stacked together. Geometrically-wise, we can imagine these bands as frustums of a circular cone.
Divide the surface area into \(n\) frustum portions.
Solve for the surface area of each division.
Combine the results to get an approximate result.
The surface area of the frustum \(S\) is equal to \(2πrL\) where \(r\) is the average radius between the top and bottom base and \(L\) is the slant length.
Our answer is only an approximation. To have a better result, we increase the number of equally-divided \(n\) frustums and solve for each. We sum these up to get a more precise answer.
As we increase the number of n frustums, our approximation is better.
As the number of frustum increases, it gets smaller.
The smaller the frustum, its area \(A\) and slant length \(L\) become smaller.
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Building from our approximate solution, we can solve the actual surface area of the solid. We will only need to modify two things in the approximate solution:
The area of the band (frustum) must be so small that we represent it with a differential \(dA\)
We need an operation that will sum up all of these tiny divisions
We can combine all \(dA\) through integration; hence, the general equation to solve for the surface area \(S\), between two points along the function is:
\(S=\int_{a}^{b}{dA}\)
\(S\) is the surface area
\(dA\) is the differential area
\(a\) is the lower limit position (endpoint with the lowest value)
\(b\) is the upper limit position (endpoint with the highest value)
Differential Frustum
The differential frustum \(dA\) is the division we use to sum all areas along the solid's surface. It results from having a differential slant length \(dL\). From its elementary formula, we can express it as:
\(S=2 \pi \int_{a}^{b} r(x) \cdot dL\)
\(S\) is the surface area
\(\pi\) is mathematical constant pi
\(r(x)\) is the average radius expressed as a function
\(dL\) is the differential length
\(a\) is the lower limit position (endpoint with the lowest value)
\(b\) is the upper limit position (endpoint with the highest value)
We can also think of the radius function \(r(x)\) as the slant boundary of the solid. Since we have a differential frustum, we can express it directly as it is and not the average of its top and base radii (unlike in Elementary geometry).
Differential Slant Length
To get the differential slant length \(dL\), we form a right triangle with its differential legs \(dx\) and \(dy\). We can isolate \(dL\), similar to getting the arc of a curve using integration. As a result, \(dL\) can either be:
Along x: \(d L=\sqrt{1+\left(\frac{d y}{d x}\right)^2} dx\)
Along y: \(d L=\sqrt{1+\left(\frac{d x}{d y}\right)^2} dy\)
From this, the frustum element's differential slant length \(dL\) will vary depending on the orientation.
Since there are two forms, the radius function \(r(x)\) must correspond properly to \(dL\). It must be in terms of the differential variable:
If the radius function is in terms of \(x\), we use \(d L=\sqrt{1+\left(\frac{d y}{d x}\right)^2} dx\)
If the radius function is in terms of \(y\), we use \(d L=\sqrt{1+\left(\frac{d x}{d y}\right)^2} dy\)
As a result, there are two specific expressions for the surface area:
Solving the surface area of a solid of revolution by integration expands on the approximation solution.
The logic is to subdivide the area needed with frustum bands, solve for the surface area of each part, and add all of these areas.
It is necessary to keep the surface area small to increase the accuracy of the solution; hence, we must represent the parts as the differential \(dA\).
We can combine all \(dA\) through integration; hence, the general equation to solve for the surface area between two points: \(S=\int_{a}^{b}{dA}\)
The differential frustum \(dA\) is the division we use to sum all areas along the solid's surface: \(S=2 \pi \int_{a}^{b} r(x) \cdot d L\). \(S\) is the surface area, \(\pi\) is mathematical constant pi, \(r(x)\) is the average radius expressed as a function, \(dL\) is the differential length, \(a\) is the lower limit position (endpoint with the lowest value), \(b\) is the upper limit position (endpoint with the highest value)
Two ways exist to express the differential slant length \(dL\). It is similar to getting the arc length of a curve using integration.
In terms along the x-axis, \(S\) is equal to: \(S=2 \pi \int_{a}^{b} r(x) \cdot \sqrt{1+\left(\frac{dy}{dx}\right)^2} dx\)
In terms along the y-axis, \(S\) is equal to: \(S=2 \pi \int_{a}^{b} r(y) \cdot \sqrt{1+\left(\frac{dx}{dy}\right)^2} dy\)
