Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 5: Problem 418
The integral \(\int_{0}^{2 \pi} \int_{0}^{1} \int_{r}^{1} d z d r d \theta\) represents the volume of a right cone.
Short Answer
Expert verified
The volume of the right cone is \( \pi \).
Step by step solution
01
Understand the Problem
We have a triple integral \( \int_{0}^{2 \pi} \int_{0}^{1} \int_{r}^{1} dz \ dr \ d\theta \) which is used to calculate the volume. The variables \( r, z, \theta \) are likely in cylindrical coordinates, where \( r \) is the radial coordinate, \( z \) is the height, and \( \theta \) is the angle.
02
Identify the Limits of Integration
The triple integral \( \int_{0}^{2 \pi} \int_{0}^{1} \int_{r}^{1} dz \ dr \ d\theta \) involves the limits: \( r \) ranging from 0 to 1, \( z \) going from \( r \) to 1, and \( \theta \) from 0 to \( 2\pi \). This indicates a cone where the upper surface is described by \( z = r \), and \( z \) ends at 1.
03
Evaluate the Innermost Integral
The innermost integral \( \int_{r}^{1} dz \) is evaluated first. It is simply the antiderivative of 1 over the interval \( [r, 1] \), yielding \( z \Big|_{r}^{1} = 1 - r \).
04
Evaluate the Middle Integral
Next, integrate \( 1 - r \) with respect to \( r \) over \( [0, 1] \). This gives \( \int_{0}^{1} (1-r) dr = \left[r - \frac{r^2}{2} \right]_{0}^{1} = 1 - \frac{1}{2} = \frac{1}{2} \).
05
Evaluate the Outermost Integral
Finally, integrate \( \frac{1}{2} \) with respect to \( \theta \) over \( [0, 2\pi] \). This results in \( \int_{0}^{2\pi} \frac{1}{2} d\theta = \frac{1}{2} \cdot \theta \Big|_{0}^{2\pi} = \frac{1}{2} (2\pi) = \pi \).
06
Interpret the Result
The computed volume, \( \pi \), represents the volume of the cone described by the given coordinates. In the context of the cone, \( z = 1 \) is the height, and \( r = 1 \) is the radius at the base.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cylindrical Coordinates
Cylindrical coordinates are a three-dimensional coordinate system that extends polar coordinates by adding a vertical axis. This system is perfectly suited for problems involving cylindrical shapes, like cones. By using cylindrical coordinates, we can describe points in space with three parameters:
- \( r \) for the radial distance from the origin to the point. This is similar to the radius in polar coordinates.
- \( \theta \) for the angular coordinate, which measures the angle in the \( xy \)-plane from the positive \( x \)-axis.
- \( z \) for the height or the vertical coordinate along the z-axis.
Volume of a Cone
The volume of a cone can be calculated using triple integrals, particularly suited to cylindrical coordinates. A cone is a solid with a circular base that tapers smoothly to a point, known as the apex.
In the exercise, the cone's volume is defined between certain limits using the triple integral:$$ \int_{0}^{2 \pi} \int_{0}^{1} \int_{r}^{1} dz \ dr \ d\theta $$The integration order reflects the natural cylindrical shape of the cone. Here are the parameters that define the cone:
In the exercise, the cone's volume is defined between certain limits using the triple integral:$$ \int_{0}^{2 \pi} \int_{0}^{1} \int_{r}^{1} dz \ dr \ d\theta $$The integration order reflects the natural cylindrical shape of the cone. Here are the parameters that define the cone:
- The base's radius: \( r = 1 \).
- Height: \( z = 1 \), the maximum value of \( z \), where the cone reaches its apex.
- The surface slope follows the equation \( z = r \).
Limits of Integration
Limits of integration are crucial in defining the boundaries of a region over which an integral is evaluated, especially in multiple integrals. In this exercise, we calculate the volume of a cone by carefully setting the limits in cylindrical coordinates:
- \( r \) ranges from 0 to 1. This defines the circular slice of the cone's base, starting from the center to the edge.
- \( \theta \) is from 0 to \( 2\pi \), indicating a full rotation around the circle, capturing all angles of the circular base.
- \( z \) depends on \( r \), varying from \( r \) to 1. This constraint creates the cone shape, where \( z \) increases linearly with \( r \), up to its maximum height which is 1.
