Finding the Integral Limits for an Inverted Cone Body

However, the problem specifies that we are only integrating over the part of the cylinder that satisfies x^2 + y^2 ≤ z^2, which is a cone with a base radius of 1 and height of 1. Therefore, the limits of integration must reflect this restriction, hence p ≤ z.
  • #1
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Homework Statement




The problem is attached in the picture.

The Attempt at a Solution



My question is, why can't the limits of ∫(dp) be from 0 to 1?

The shape is of the body is an inverted cone of angle 45 degree. Then isn't the question simply to find the integral of that function from p = 0 to p = 1, z = 0 to z = 1 and ∅ = 0 to ∅ = 2∏?

Since:

x = p cos∅
y = p sin∅
z = z


x2+y2 ≤ z2, 0 ≤ z ≤ 1

p2 ≤ z2

p ≤ z
 

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  • #2
unscientific said:

Homework Statement



The problem is attached in the picture.

The Attempt at a Solution



My question is, why can't the limits of ∫(dp) be from 0 to 1?

The shape is of the body is an inverted cone of angle 45 degree. Then isn't the question simply to find the integral of that function from p = 0 to p = 1, z = 0 to z = 1 and ∅ = 0 to ∅ = 2∏?
Doing what you suggest would be integrating over a right circular cylinder of radius 1 and height of 1.
 

1. What is the definition of a limit of an integral?

The limit of an integral is a mathematical concept that represents the value that a function approaches as the input variable approaches a specific value. It is denoted by the notation lim and is used to describe the behavior of a function near a particular point.

2. How do you calculate the limit of an integral?

To calculate the limit of an integral, you first need to evaluate the integral at the specific point in question. Then, you need to evaluate the function as the input variable approaches the given value. If the function approaches a finite value, then that value is the limit. If the function approaches infinity or negative infinity, then the limit does not exist.

3. Why is the limit of an integral important in mathematics?

The limit of an integral is important in mathematics as it allows us to study the behavior of a function near a specific point. It helps us understand the continuity and differentiability of a function, as well as its behavior at critical points. It is also essential in solving various real-world problems and in the development of advanced mathematical concepts.

4. Can the limit of an integral be calculated using different methods?

Yes, there are different methods for calculating the limit of an integral, such as using L'Hospital's rule, the squeeze theorem, or substitution. The method used will depend on the function and the specific point in question.

5. Are there any real-life applications of the limit of an integral?

Yes, the limit of an integral has many real-life applications, such as in physics, engineering, and economics. It is used to determine the maximum and minimum values of a function, calculate areas and volumes, and model various real-world phenomena, such as population growth and chemical reactions.

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