Triple Integrals: Finding Limits Without Sketching

In summary, the conversation discusses finding the limits of integration for a triple integral without sketching the region. One person suggests using equations to determine the limits, while the other questions the difficulty of sketching a plane.
  • #1
roam
1,271
12

Homework Statement



Here is a solved problem:

[PLAIN]http://img3.imageshack.us/img3/6948/97765276.gif

In part (e), they formulated the triple integral using the limits of integration they found by sketching the region. Is there a way we can find the limits of integration without sketching the region? To me, sketching this graph looks complicated, so I'm wondering if we can determine the limits for the iterated integral by just using the given equations...

The Attempt at a Solution



I appreciate any help. :smile:
 
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  • #2
Why would sketching a plane be complicated? 3 points are enough to determine a plane. So, for example, if you have some plane given with x + y + z = C, set x = y = 0 and find z, then set y = z = 0, and find x, and set x = z = 0, and find y.
 

Related to Triple Integrals: Finding Limits Without Sketching

1. What are triple integrals and why are they important in science?

Triple integrals are a type of mathematical operation used to calculate the volume of three-dimensional objects or regions. They are important in science because many physical and natural phenomena can be modeled and analyzed using three-dimensional shapes, making triple integrals a valuable tool for understanding and predicting these phenomena.

2. How do you find the limits for a triple integral without sketching?

To find the limits for a triple integral without sketching, you can use the method of slicing, which involves breaking down the three-dimensional region into smaller, simpler shapes and finding the limits of integration for each shape. This method can be more efficient and accurate than sketching, especially for complex regions.

3. What is the difference between a definite and indefinite triple integral?

A definite triple integral is used to calculate the exact value of a triple integral over a specific region, while an indefinite triple integral is used to find the general antiderivative of a function over a three-dimensional space. Definite triple integrals have fixed limits of integration, while indefinite triple integrals have variables as their limits.

4. What are some real-life applications of triple integrals?

Triple integrals have many real-life applications in fields such as physics, engineering, and economics. They can be used to calculate the volume of a solid object, the mass of a three-dimensional object, the center of mass of an object, and the moment of inertia of an object. They are also used in probability and statistics to calculate the probability of events occurring in three-dimensional space.

5. How can I improve my understanding and skills in solving triple integrals?

To improve your understanding and skills in solving triple integrals, it is important to practice regularly and familiarize yourself with the different methods and techniques used to solve them. You can also seek assistance from textbooks, online resources, or a tutor to help you with any difficult concepts. Additionally, developing a strong foundation in calculus and geometry can also greatly improve your ability to solve triple integrals.

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