Surface area

Definition

Surface area is the measure of the total area that the surface of an object occupies. In calculus, it often involves integrating to find the area of a surface generated by rotating a curve around an axis.

5 Must Know Facts For Your Next Test

1. The formula for surface area of revolution around the x-axis is $2\pi \int_a^b f(x) \sqrt{1+(f'(x))^2} \, dx$.
2. For surfaces generated by rotating around the y-axis, the formula is $2\pi \int_a^b x \sqrt{1+(f'(x))^2} \, dx$.
3. To calculate surface area using parametric equations, use $2\pi \int_a^b g(t) \sqrt{(f'(t))^2 + (g'(t))^2} \, dt$ for rotation around the x-axis.
4. Arc length must be calculated before determining surface area, as it forms part of the integrand in these formulas.
5. When dealing with polar coordinates, the formula for surface area involves integrating $r(\theta)$ and its derivative.

Review Questions

• What is the integral formula for finding the surface area when revolving a curve around the x-axis?
• How does one modify the surface area formula when dealing with parametric equations?
• Why is calculating arc length important before finding surface area?

Related terms

Arc Length: The distance along a curve between two points, calculated using integration.

Parametric Equations: Equations that express coordinates as functions of a parameter, often used to describe curves.

Polar Coordinates: A coordinate system where each point on a plane is determined by an angle and a distance from a reference point.