Arc Length of a Curve and Surface Area

Arc length is the total length of a curve between two points. An arc length is the whole length of a curve, not the straight line from one point to another.
Divide the curve into tiny parts and measure them independently to find the arc length. Add all these segments to find the curve's length. Calculus and geometry require this approach to solve curve shapes and pathways. The area of a surface of revolution is the area gained by rotating a curve around a fixed axis. This notion is fundamental to calculus and has implications in physics, engineering, and other domains where knowing the properties of three-dimensional forms is critical.

In this article, we will understand how to calculate the arc length of the curve for functions x and y and the area of a surface of revolution.

What is Arc Length of a Curve?

The arc length of a curve is defined as the distance between two points on a continuous curve. It is essentially the "length" of the route drawn by the curve. Calculating the arc length is simple for simple geometric shapes such as circles, but it becomes more complicated for arbitrary curves specified by mathematical functions.

Arc Length of the Curve y = f(x)

To find the arc length of a curve defined by the function f(x) from x=a to x=b, we follow these steps:

Step 1: Divide the interval [a,b] into n subintervals of equal width, Δx = (b-a)/n​.

Step 2: For each subinterval [x_i−1​,x_i​], calculate the change in vertical distance Δyi​=f(x_i​)-f(x_i−1​).

Step 3: Use the Pythagorean theorem to calculate the length of each line segment: \sqrt{(\Delta x)^2 + (\Delta y_i)^2}​.

Step 4: Apply the Mean Value Theorem to find a point x_i^*​ in each subinterval where f'(x_i^*) = \frac{\Delta y_i}{\Delta x}.

Step 5: Rewrite the length of each line segment as \sqrt{1 + [f'(x_i^*)]^2} \cdot \Delta x.

Step 6: Sum the lengths of all line segments using a Riemann sum: L \approx \sum_{i=1}^{n} \sqrt{1 + [f'(x_i^*)]^2} \cdot \Delta x

Step 7: As n approaches infinity, the Riemann sum converges to the actual arc length L: L = \lim_{n \to \infty} \sum_{i=1}^{n} \sqrt{1 + [f'(x_i^*)]^2} \cdot \Delta x.

Step 8: Therefore, the arc length of the curve from x=a to x=b is given by the integral: L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} \, dx.

Formula of Arc Length for y = f(x)

Let f(x) be a continuous function defined on a closed interval [a,b], where a and b are real numbers. The arc length L of the curve represented by y = f(x) from x = a to x = b is given by:

L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx

where,

• dy/dx​ represents the derivative of f(x) for x, i.e., the slope of the curve at any point x.
• The integral is taken over the interval [a,b] that defines the segment of the curve for which the arc length is to be calculated.

Arc Length of the Curve x = g(y)

The arc length of a curve x = g(y) has a similar process to that we just explained for y=f(x).

To calculate the length of the curve between two points: (g(c),c) and (g(d),d), divide the interval [c,d] into smaller segments using a regular partition. For each segment, calculate the change in vertical distance, denoted as Δy, and the change in horizontal distance, denoted as Δx_i ​= g(y_i​) - g(y_i−1​).

These values will be used in calculating the length of each line segment using the Pythagorean theorem: \sqrt{(\Delta x_i)^2 + (\Delta y)^2}.

Formula of Arc Length for x = g(y)

Given c and d are real values, let g(y) be a smooth function defined over a closed interval [c,d]. After then, the following will provide the length of the curve denoted by x = g(y) from the points (g(c),c) to (g(d),d):

L = \int_{c}^{d} \sqrt{1 + \left(\frac{dx}{dy}\right)^2} \, dy

here,

• dx/dy represents the derivative of g(y) with respect to y. This shows the slope of the curve at any point y.
• The integral is taken over the interval [c,d] that defines the segment of the curve for which the arc length is to be calculated.

Area of a Surface of Revolution

When a curve is revolved around an axis, it forms a three-dimensional surface. The surface area of the created shape is known as the surface area of revolution. The computation involves integrating to account for the continuous character of the surface generated by rotation.

Methods are used to find surface area of a curve spun around the x-axis is mentioned below:

• Split [a, b] into smaller intervals.
• To estimate the surface area, find the surface areas of simpler forms.
• With line segments, first approximate the curve. Rotate these line segments about the x-axis to form bands. Every frustum resembles a cone.
• We require the slant height (l) and top and bottom radii (r₁ and r₂) of a frustum in order to compute its surface area.
• Cones have lateral surface areas of πrs, where s is the slant height and r is the base radius.

NOTE: A frustum is a portion of a cone, hence its lateral surface area is the lateral surface area of the entire cone less the lateral surface area of the smaller cone (the pointed tip) that was removed.

Therefore, we calculate the slant height of the frustum using the formula : \frac{{r_2}}{{r_1}} = \frac {s - l}s

r₂s = r₁ (s-l)

r₂s = r₁s - r₁l

r₁l = r₁s - r₂s

r₁l = (r₁ - r₂) s

\frac{{r_1 l}}{{r_1 - r_2}} = s

The difference in lateral surface area between the smaller cone and the full cone gives the lateral surface area (S) of the frustum.

Lateral Surface Area of the Whole Cone = πr_1​s

Lateral Surface Area of the Smaller Cone = πr_2​(s−l)

S = πr₁​s - πr₂​(s−l)

substituting the values of s in the formula we get,

S = \pi r_1 \left(\frac{r_1 (s - l)}{r_1 - r_2}\right) - \pi r_2 \left(\frac{r_1 (s - l)}{r_1 - r_2} - l\right)

S = \pi \left(\frac{r_1^2 (s - l)}{r_1 - r_2}\right) - \pi \left(\frac{r_2 (r_1 (s - l) - l (r_1 - r_2))}{r_1 - r_2}\right)

S = \pi \left(\frac{r_1^2 (s - l) - r_2 (r_1 (s - l) - l (r_1 - r_2))}{r_1 - r_2}\right)

S = \pi \left(\frac{r_1^2 s - r_1^2 l - r_1 r_2 s + r_1 l - r_2 r_1 s + l r_2 + r_1 l - l r_2}{r_1 - r_2}\right)

S = \pi \left(\frac{r_1^2 s - r_1 r_2 s - r_1 r_2 s + r_2^2 s}{r_1 - r_2}\right)

S = \pi \left(\frac{(r_1^2 - r_1 r_2 - r_1 r_2 + r_2^2) s}{r_1 - r_2}\right)

S = \pi \left(\frac{(r_1^2 - 2r_1 r_2 + r_2^2) s}{r_1 - r_2}\right)

S = \pi \left(\frac{(r_1 - r_2)^2 s}{r_1 - r_2}\right)

S = \pi (r_1 - r_2) s

To calculate the surface area of each band formed by revolving the line segments around the x-axis, we use the formula:

S = π(r1​+r2​)l

In this case, r1​+r2​ corresponds to f(x_i−1​) + f(x_i​), and l is simply Δx, the length of the line segment used to generate the frustum.

So, substituting these values into the formula, we get,

S = \pi(f(x_{i-1}) + f(x_i)) \frac{\Delta x}{2} \sqrt{1 + \left(\frac{\Delta y_i}{\Delta x}\right)^2}

Using Mean Value Theorem, we get,

S = \pi(f(x_{i-1}) + f(x_i)) \frac{\Delta x}{2} \sqrt{1 + (f'(x^*_i))^2}

Furthermore, by the Intermediate Value Theorem, there is a point x^{**}_i such that f(x^{**}_i ) = \frac{1}{2}[f(x_{i-1}) + f(x_i)] , giving us:

S = 2\pi f(x^{**}_i) \frac{\Delta x}{2} \sqrt{1 + (f'(x^*_i))^2}

The approximate surface area of the entire surface of revolution is then given by the sum:

\text{Surface Area} \approx \sum_{i=1}^{n} 2\pi f(x^{**}_i) \frac{\Delta x}{\sqrt{1 + (f'(x^*_i))^2}}

Taking the limit as ( n ) approaches infinity, we get:

\text{Surface Area} = \lim_{n \to \infty} \sum_{i=1}^{n} 2\pi f(x^{**}_i) \frac{\Delta x}{\sqrt{1 + (f'(x^*_i))^2}}

\text{Surface Area} = \int_{a}^{b} 2\pi f(x) \sqrt{1 + (f'(x))^2} \, dx

Also, Check

• Area Between Two Curves
• Area Under The Curve
• Area between Polar Curves