Triple Integrals: Solving G Bounded by x, 2-x, and y^2

In summary, to solve the triple integral over G, where G is bounded by z = x, z = 2-x, and z = y^2, the lower bound for x is 1 and the upper bound is 2. The lower bound for z is 0 and the upper bound is y^2. The lower bound for y is the square root of x and the upper bound is the square root of (2-x). However, integrating in this case may be difficult.
  • #1
stunner5000pt
1,461
2
[tex] \int \int \int_{G} (xy + xz) dx dy dz [/tex]
G bounded by z=x, z= 2-x, and z = y^2

solving the first 2 i get x =1

equating y^2 = z =x and y^2 = 2-x
so x can go from 1 to 2?
not sure how to proceed for the y part, however..

please helppppp
 
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  • #2
so far i got the x lower boun to be 1 and upper boun to be 2

i (think) that the z lower bound is zero and upper bound is y^2

also the y lower boun would be root x and upper bound would be root (2-x)


is this correct??
Integrating would be a real bugger in this case

please help!
 
  • #3
Try integrating dx first, with z constant. I've attached a graph to help you figure out the limits. Set your constraints in terms of x= and y=

http://www.public.asu.edu/~hyousif/prob2.JPG"

The thin slant is z=2-x, the plane is z=x and the parabola is obviously z=y^2
 
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Related to Triple Integrals: Solving G Bounded by x, 2-x, and y^2

1. What is a triple integral?

A triple integral is a type of mathematical calculation used in multivariable calculus to find the volume of a three-dimensional shape. It involves integrating a function over a region in three-dimensional space.

2. How do you set up a triple integral?

To set up a triple integral, you first need to define the limits of integration for each variable. This is done by identifying the boundaries of the region in three-dimensional space. Then, you write the function to be integrated and the differentials of each variable.

3. What is the process for solving a triple integral?

The process for solving a triple integral involves first setting up the integral, then evaluating it using techniques such as substitution, partial fractions, and trigonometric identities. Finally, you solve the resulting integral to find the volume of the bounded region.

4. How do you determine the boundaries for a triple integral?

The boundaries for a triple integral can be determined by visualizing the bounded region and identifying the range of values for each variable. They can also be found by solving equations that represent the boundaries of the region.

5. Can you use triple integrals to find other quantities besides volume?

Yes, triple integrals can also be used to find other quantities such as mass, center of mass, and moments of inertia. In these cases, the function being integrated represents a physical property and the limits of integration are determined by the shape and density of the object.

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