How to find surface area using integration?

You can find the surface area of a solid formed by revolving a curve around an axis using integration by applying a specific formula derived from summing infinitesimally small surface area segments.

When you revolve a curve defined by a function, say $y=f(x)$, around an axis (like the x-axis) over an interval $[a, b]$, you create a three-dimensional solid known as a solid of revolution. The surface area of this solid (excluding the ends, if they exist) can be calculated using an integral.

The Concept

Imagine dividing the curve into many tiny segments. When each segment is revolved around the axis, it forms a thin band or frustum of a cone. The surface area of the solid of revolution is the sum of the areas of all these infinitesimally thin bands. Integration provides the tool to perform this infinite summation precisely.

The Formula

The surface area (often denoted as $S$) of a solid generated by revolving a curve $y = f(x)$ around the x-axis from $x=a$ to $x=b$ is given by the formula:

$$
\text{Surface Area} = \lim{n\to\infty} \sum{i=1}^n 2\pi f(x_i^{*}) \Delta x \sqrt{1+(f'(x_i^))^2} = \int_{a}^{b} \left(2\pi f(x)\sqrt{1+(f'(x))^2}\right)dx
$$

This formula directly comes from the reference provided.

Breaking Down the Formula

Let's understand the components of the integral:

• 2πf(x): This represents the circumference of a circular cross-section of the solid at a given x-value. Think of f(x) as the radius of this circle when revolving around the x-axis.
• f'(x): This is the derivative of the function f(x) with respect to x. It represents the slope of the tangent line to the curve at any point x.
• √1+(f'(x))²: This term comes from the arc length formula, ds = √1 + (dy/dx)² dx. When multiplied by dx, √1+(f'(x))² dx represents an infinitesimally small segment of the arc length of the curve y=f(x).
• 2πf(x)√1+(f'(x))² dx: This entire expression inside the integral represents the surface area of one infinitesimally thin band formed by revolving a small arc length ds around the axis. It's essentially circumference × slant height (ds).
• ∫ab (...) dx: This is the definite integral from x=a to x=b. It sums up the areas of all the infinitesimally thin bands along the curve from the start point a to the end point b to give the total surface area.

Steps to Find Surface Area Using Integration

To calculate the surface area of a solid of revolution using this method:

1. Identify the Function and Interval: Determine the function $y=f(x)$ that defines the curve being revolved and the interval $[a, b]$ over which the revolution occurs. Also, identify the axis of revolution (e.g., x-axis or y-axis). The formula provided is for revolution around the x-axis.
2. Calculate the Derivative: Find the first derivative of the function, $f'(x)$.
3. Set up the Integral: Substitute $f(x)$ and $f'(x)$ into the formula:
   $$
   S = \int{a}^{b} 2\pi f(x)\sqrt{1+(f'(x))^2} dx
   $$
   *Note: If revolving around the y-axis, the formula changes. If your curve is $x=g(y)$ revolved around the y-axis from $y=c$ to $y=d$, the formula is $S = \int{c}^{d} 2\pi g(y)\sqrt{1+(g'(y))^2} dy$. If your curve is $y=f(x)$ revolved around the y-axis, you'd use $x$ as the radius and express $dx$ in terms of $dy$ from the arc length, often using parameterization or rewriting the integral with respect to x: $S = \int_{a}^{b} 2\pi x\sqrt{1+(f'(x))^2} dx$ (assuming $x \ge 0$ on $[a,b]$).*
4. Evaluate the Integral: Calculate the definite integral to find the numerical value of the surface area. This step often requires techniques of integration, such as substitution or trigonometric substitution.

By following these steps and using the correct formula based on the axis of revolution and how the curve is defined, you can accurately calculate the surface area of a solid of revolution.