Find the volume using shell and disk method

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SUMMARY

The discussion centers on calculating the volume of a solid using both the shell and disk methods, specifically focusing on the expression $\displaystyle V = 18\pi - 2\pi \int_0^{\pi/2} x \cos{x} \, dx$. The value of $18\pi$ is derived from the volume of a hemisphere with radius $r = 3$, calculated using the formula $\dfrac{2\pi r^3}{3}$. Participants seek clarification on the origin of the $18\pi$ term in the volume equation.

PREREQUISITES
  • Understanding of integral calculus, specifically volume calculations using the shell and disk methods.
  • Familiarity with trigonometric functions and their integrals, particularly $\cos{x}$ and $\arccos{y}$.
  • Knowledge of the formula for the volume of a hemisphere, $\dfrac{2\pi r^3}{3}$.
  • Ability to evaluate definite integrals and interpret their geometric significance.
NEXT STEPS
  • Study the shell method for volume calculation in detail.
  • Learn about the disk method and its applications in volume problems.
  • Practice evaluating integrals involving trigonometric functions, especially $\int x \cos{x} \, dx$.
  • Explore the geometric interpretation of volumes derived from integrals.
USEFUL FOR

Students and educators in calculus, mathematicians focusing on volume calculations, and anyone interested in mastering integral techniques for finding volumes of solids.

jaychay
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Can you please help me ?
I have tried to do it but I end up getting the wrong answer.

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$\displaystyle V = 18\pi - 2\pi \int_0^{\pi/2} x \cos{x} \, dx$

$\displaystyle V = 18\pi - \pi \int_0^1 [\arccos{y}]^2 \, dy$
 
Last edited by a moderator:
skeeter said:
$\displaystyle V = 18\pi - 2\pi \int_0^{\pi/2} x \cos{x} \, dx$

$\displaystyle V = 18\pi - \pi \int_0^1 [\arccos{y}]^2 \, dy$
Can you tell me where did 18 pi come from ?
 
jaychay said:
Can you tell me where did 18 pi come from ?

volume of a hemisphere is $\dfrac{2\pi r^3}{3}$ and $r = 3$
 

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