10.2: Calculus with Parametric Equations

Question

Find the surface area generated by rotating the given curve about the y-axis.
x = 9t^2, y = 6t^3, 0 ≤ t ≤ 5

 

 

Answer 1

The following is not correct because it is for volume not surface area! Look up the formula for surface area for a solid of revolution and follow a similar pattern.

 

Normally a volume of revolution rotated about y-axis would be calculated by pi∫ from y₁ to y₂ (x² dy).

 

Since x = 9t² and y = 6t³ , use those value in that formula.

 

pi ∫ 0 to 5 (9t²)² 18t² dt   (where dy/dt = 18t²)

 

For surface area about y-axis where f(x) = 9t² , g(x) = 6t^3.

 

S= 2pi ∫ x √(f'(t)² + g'(t)²) dt where limits of integratiion are 0 to 5.