Last updated on August 5th, 2025
The surface area of revolution refers to the total area of the surface generated when a curve is revolved around a line, typically one of the coordinate axes. This concept is crucial in calculus and geometry, providing insights into the properties of 3-dimensional shapes formed by rotating a 2-dimensional curve. In this article, we will explore the surface area of revolution.
The surface area of revolution is the total area of the surface formed by rotating a curve around a given axis. It is measured in square units.
A common example is generating a 3D shape by rotating a 2D line or curve around the x-axis or y-axis. The resulting shape can include common geometric figures like cylinders, cones, and spheres.
The surface area is calculated using integral calculus to account for the curve's continuous nature being revolved.
The formula for calculating the surface area of revolution depends on the axis of rotation.
For a curve y = f(x) revolved around the x-axis from x = a to x = b, the formula is: Surface Area = ∫[a to b] 2πy√(1+(dy/dx)2) dx
Similarly, for a curve x = g(y) revolved around the y-axis from y = c to y = d, the formula is: Surface Area = ∫[c to d] 2πx√(1+(dx/dy)2) dy
When a curve y = f(x) is revolved around the x-axis, the surface area of revolution is calculated using the formula:
Surface Area = ∫[a to b] 2πy√(1+(dy/dx)^2) dx
Here, y = f(x) is the function being revolved, and dy/dx is its derivative with respect to x.
This formula accounts for the circular cross-sections formed by the rotation.
For a curve x = g(y) revolved around the y-axis, the surface area is determined by: Surface Area = ∫[c to d] 2πx√(1+(dx/dy)^2) dy
In this scenario, x = g(y) is the function being rotated, and dx/dy is its derivative with respect to y.
The formula considers the horizontal cross-sections made by the rotation.
The volume of a solid of revolution is the total space occupied by the 3D shape formed when a curve is revolved around an axis. It is calculated using the disk or washer method, depending on the shape.
For example, the volume for a curve y = f(x) about the x-axis is: Volume = π∫[a to b] (f(x))^2 dx
Students might confuse whether the curve is rotated about the x-axis or y-axis, leading to incorrect formula usage. Ensure clarity on the axis before applying the formula.
Given y = x^2, dy/dx = 2x. Surface Area = ∫[0 to 2] 2π(x^2)√(1+(2x)^2) dx = ∫[0 to 2] 2π(x^2)√(1+4x^2) dx Calculating the integral gives approximately 25.13 square units.
Find the surface area of revolution for x = y^2 rotated about the y-axis from y = 0 to y = 1.
Surface Area = 19.74 square units
Given x = y² Surface area of revolution Surface area of revolution, dx/dy = 2y. Surface Area = ∫[0 to 1] 2πy²√(1+(2y)²) dy = ∫[0 to 1] 2πy²√(1+4y²) dy Calculating the integral gives approximately 19.74 square units.
Find the surface area of revolution for y = √x rotated about the x-axis from x = 1 to x = 4.
Surface Area = 13.68 square units
Given y = √x, dy/dx = 1/(2√x). Surface Area = ∫[1 to 4] 2π√x√(1+(1/(2√x))²) dx = ∫[1 to 4] 2π√x√(1+1/(4x)) dx Calculating the integral gives approximately 13.68 square units.
Find the surface area of revolution for x = 1/y rotated about the y-axis from y = 1 to y = 3.
Surface Area = 10.47 square units
Given x = 1/y, dx/dy = -1/y². Surface Area = ∫[1 to 3] 2π(1/y)√(1+(-1/y²)²) dy = ∫[1 to 3] 2π(1/y)√(1+1/y⁴) dy Calculating the integral gives approximately 10.47 square units.
Find the surface area of revolution for y = 3x rotated about the x-axis from x = 0 to x = 1.
Surface Area = 59.22 square units
Students often encounter challenges in calculating the surface area of revolution due to errors in applying formulas or integrating. Below are some common mistakes and how to avoid them.
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