How to Rotate a Surface Area about the Y-Axis?

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SUMMARY

The discussion focuses on calculating the surface area generated by rotating the curve defined by the equation y = x²/4 - ln(x)/2 around the y-axis for the interval 1 ≤ x ≤ 2. The user successfully applied the formula for rotation about the x-axis using the integral 2π * ∫(f(x) * sqrt(1 + (f'(x))²)) dx, where f'(x) = x/2 - 1/2x. However, the challenge lies in rearranging the equation to express x in terms of y for the y-axis rotation, which requires further exploration of implicit functions or inverse functions.

PREREQUISITES
  • Understanding of integral calculus, specifically surface area calculations.
  • Familiarity with the concept of rotating curves about axes.
  • Knowledge of implicit functions and their rearrangement.
  • Basic differentiation techniques to find f'(x).
NEXT STEPS
  • Study the method for calculating surface area of revolution around the y-axis.
  • Learn how to rearrange equations to express x in terms of y for implicit functions.
  • Explore the application of the inverse function theorem in calculus.
  • Practice problems involving surface area calculations for various curves.
USEFUL FOR

Students and educators in calculus, particularly those focusing on surface area of revolution, as well as anyone seeking to deepen their understanding of implicit functions and their applications in real-world scenarios.

Tarpie
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Greetings,

y=x2/4 - ln(x)/2 from 1=<x<=2

rotated about the y-axis.

I did the equation rotating about the x-axis via 2pi* integral (f(x)*sqrt(1+f'(x)^2)) dx
with dy/dx = x/2 - 1/2x

but the question calls for rotation about y and i can't seem to rearrange the equation to isolate for x in terms of y.

Any ideas?

Thanks
 
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