How do you find the surface area of a solid of revolution?

Question

Isabella Ackerman · Accepted Answer

If the solid is obtained by rotating the graph of #y=f(x)# from #x=a# to #x=b#, then the surface area #S# can be found by the integral #S=2pi int_a^b f(x)sqrt{1+[f'(x)]^2}dx#

Luke Alvarez · Answer

undefined To find the surface area of a solid of revolution, you can use the formula for surface area of revolution, which is given by:

Surface Area = 2π ∫[a, b] f(x) √(1 + [f'(x)]^2) dx

Where:
- f(x) represents the function that generates the solid of revolution.
- [a, b] denotes the interval over which the function is defined and rotated.
- f'(x) is the derivative of the function with respect to x.

This formula integrates the circumference of infinitesimally thin circles (2πf(x)) along the length of the solid, taking into account the curvature of the function through the square root term. The integral sums up these circumferences over the interval [a, b], resulting in the total surface area of the solid of revolution.

Isaiah Allen · Answer

undefined To find the surface area of a solid of revolution generated by revolving a curve around a given axis:

1. Determine the function representing the curve.
2. Identify the interval over which the curve is revolved.
3. Use the formula for the surface area of revolution:

Surface Area = ∫[a, b] 2π f(x) √(1 + (f'(x))^2) dx

Where:
   - f(x) is the function representing the curve.
   - f'(x) is the derivative of the function.
   - a and b are the limits of integration corresponding to the interval of revolution.

4. Integrate the expression obtained from the formula over the given interval [a, b].
5. Evaluate the integral to find the surface area.