What is the range for x in a triple integral for a wedge in the first octant?

In summary, the triple integral to represent the volume of the solid wedge in the first octant and from the cylinder y^2 + z^2 <= 1 by the planes y=x, x=0, z=0 is: ∭f(x,y,z) dV, where the bounds for z are 0 ≤ z ≤ √(1-y^2), the bounds for y are 0 ≤ y ≤ 1, and the bounds for x depend on the placement of the solid in the integration and may be easier to calculate using cylindrical coordinates.
  • #1
naspek
181
0
Write a triple integral to represent the volume of the solid

The wedge in the first octant and from the cylinder y^2 + z^2 <= 1 by the planes
y=x, x=0, z=0

First..
i find the range for z..; 0 <=z<= sqrt(1- y^2)

then...
i find the range for y..; let z =0
0<=y<=1

next, if i let z = y = 0

how am i going to find the range for x?
 
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  • #2
have you drawn a picture? it depends where you put it in the integration...

it could be 0 to y for example...

this may be easier in cylindrical co-ords, though i haven't tried it
 

FAQ: What is the range for x in a triple integral for a wedge in the first octant?

What are the limits of integration for triple integrals?

The limits of integration for triple integrals depend on the type of region being integrated over. For rectangular regions, the limits are determined by the minimum and maximum values of x, y, and z in the region. For more complex regions, it may be necessary to split the integral into multiple parts and use different limits for each part.

How do I determine the order of integration for a triple integral?

The order of integration for a triple integral is determined by the type of region being integrated over. For simple rectangular regions, the order can be determined by considering the order of the variables in the integrand. For more complex regions, it may be necessary to draw a graph or use other methods to determine the appropriate order.

Can the limits of integration for a triple integral be negative?

Yes, the limits of integration for a triple integral can be negative. This is especially common when integrating over regions that extend into the negative x, y, or z directions. It is important to carefully consider the orientation of the region when setting up the limits of integration.

What is the significance of the limits of integration in a triple integral?

The limits of integration in a triple integral represent the boundaries of the region being integrated over. They determine the range of values that the variables x, y, and z can take on in the integral. The limits also affect the orientation of the region and the order of integration.

How do I set up a triple integral for a region in polar, cylindrical, or spherical coordinates?

The setup of a triple integral for a region in polar, cylindrical, or spherical coordinates depends on the type of region being integrated over. It is important to consider the appropriate limits of integration for each coordinate system, as well as the conversion factors needed to change the variables in the integral. It may also be necessary to use multiple triple integrals or make a change of variables to fully capture the region in the integral.

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