Surface area of revolution measures the area of the curved outer surface made when a curve is rotated around an axis. It matters in calculus because it connects geometry, arc length, and integration into one model. Engineers and scientists use it to estimate material needed for objects such as pipes, tanks, lenses, nozzles, and machine parts.
The key idea is to add up many tiny surface bands along a curve.
Key Facts
- For rotation around the x-axis, surface area is S = 2π ∫ from a to b y sqrt(1 + (dy/dx)^2) dx.
- For rotation around the y-axis, surface area is S = 2π ∫ from a to b x sqrt(1 + (dy/dx)^2) dx when x is the radius.
- The arc-length element for y = f(x) is ds = sqrt(1 + (dy/dx)^2) dx.
- A tiny surface band is approximated by dS = 2πr ds, where r is the distance from the curve to the axis of rotation.
- For x = g(y), use ds = sqrt(1 + (dx/dy)^2) dy and integrate with respect to y.
- Surface area units are square units, so an answer from x and y in meters is measured in m^2.
Vocabulary
- Surface of revolution
- A surface formed by rotating a curve around a fixed axis.
- Generating curve
- The original curve that is swept around an axis to create the surface.
- Arc-length element
- A small length along a curve, written as ds, used to build the surface area integral.
- Radius of rotation
- The distance from a point on the generating curve to the axis of rotation.
- Surface band
- A thin ring-like strip of surface area formed when a small piece of the curve rotates around the axis.
Common Mistakes to Avoid
- Using dx instead of ds, which ignores the slope of the curve. The surface band follows the curve, so its width must be arc length, not just horizontal change.
- Forgetting the factor 2πr, which leaves out the circumference of each rotating band. The integral must add circumferences times small slanted widths.
- Using y as the radius for every problem, which is wrong when the curve rotates around another axis. The radius is always the distance from the curve to the axis of rotation.
- Dropping the square root in sqrt(1 + (dy/dx)^2), which changes the arc-length element. Squaring the slope accounts for how much the curve tilts compared with the x-axis.
Practice Questions
- 1 Find the surface area formed by rotating y = 2x from x = 0 to x = 3 around the x-axis. Give your answer in exact form.
- 2 Find the surface area formed by rotating y = sqrt(x) from x = 1 to x = 4 around the x-axis. Set up the integral and evaluate it exactly or with a calculator.
- 3 A curve is rotated around the x-axis, then the same curve is rotated around the line y = -2. Explain how the radius in the surface area integral changes and why the new surface area is larger.