Section 9.5
Surface Area in Parametric
We combine the parametric arc length with the circumference .
1
The Surface Area Formula
Rotation About the x-axis
Surface area = ∫ (circumference) × (arc length element)
1
The Pieces
Circumference: (radius from x-axis is )
Arc element:
2
For y-axis Rotation
If rotating about the y-axis, swap the radius:
2
Worked Example
Sphere Surface Area
Rotate semicircle about the x-axis
Step 1: Find ds
Step 2: Set Up and Integrate
Final Answer
✓ The classic sphere surface area formula!
3
Level Up Examples
Surface Generated by an Astroid
Rotate the curve about the x-axis
Step 1: Find the Derivatives
Step 2: Compute ds
Step 3: Set Up the Integral
Step 4: Integrate (u-substitution)
Let , so :
Final Answer
The surface of revolution for the first-quadrant portion of the astroid
5
Practice Quiz
Loading...