Finding volume using triple integral

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SUMMARY

The discussion focuses on calculating the volume of a solid defined by the inequalities and equations: x² + y² > 1, x² + z² = 1, and z² + y² = 1 using triple integrals. The integrand is set to 1, and the integration order is dz dr dθ, although there is a suggestion to use dxdydz. The solution involves evaluating the function h(x,y) as a double integral and subsequently integrating g(x) over x, with limits that depend on the variables involved.

PREREQUISITES
  • Understanding of triple integrals in calculus
  • Familiarity with cylindrical coordinates
  • Knowledge of integration limits based on inequalities
  • Experience with multivariable functions
NEXT STEPS
  • Study the application of triple integrals in volume calculations
  • Learn about cylindrical coordinates and their use in integration
  • Explore the concept of changing the order of integration in multiple integrals
  • Investigate the evaluation of integrals with variable limits
USEFUL FOR

Students in calculus, mathematicians working with multivariable calculus, and educators teaching integration techniques will benefit from this discussion.

woogirl14
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1. Homework Statement

I need to find the volume of a solid formed by the following equations:
x^2+y^2 > 1
x^2+z^2 = 1
z^2 + y^2 =1


3. The Attempt at a Solution

I know that it is a triple integral and the integrand is 1.
I also know that I need to use dzrdrd[tex]\theta[/tex].
 
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i think u mean dxdydz

u need to first evaluate the h(x,y) = [tex]\int[/tex] dz f(x,y,z)

between the limits on z (which may depend on x and y ).
2. Evaluate the function g(x) given by
g(x) = [tex]\int[/tex]dyh(x, y),
between the limits on y (which may depend on x ).
3. Finally integrate g(x) over x between the limits on x.
 

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