# Cylindrical Coordinates Triple Integral -- stuck in one place

• fhot112
In summary, the given problem involves using cylindrical coordinates to evaluate a triple integral over a solid bounded by a cylinder and two planes. The bounds for the integral are: dtheta from 0 to 2pi, dr from 0 to 3, and dz from 0 to 5-rsintheta. The uncertainty lies in the argument for the dz bound. It should be noted that the bounds are on the coordinates themselves, not on the coordinate differentials.
fhot112

## Homework Statement

Use cylindrical coordinates to evaluate triple integral E (sqrt(x^2+y^2)dv where E is the solid that lies within the cylinder x^2+y^2 = 9, above the plane z=0, and below the plane z=5-y

## The Attempt at a Solution

So i just need to know how to get the bounds for the dz and i have everything else.

triple integral dtheta bound: 0 to 2pi, dr bound 0 to 3, and not sure if the dz bound is 5-rsintheta?

fhot112 said:
and not sure if the dz bound is 5-rsintheta?
What is your argument for this and what makes you uncertain of your argument?

Also, for clarity, the bounds are on the coordinates themselves, not on the coordinate differentials.

## 1. What are cylindrical coordinates?

Cylindrical coordinates are a type of coordinate system used in three-dimensional space to describe the position of a point. They consist of three components: the distance from the origin to the point (r), the angle between the projection of the point onto the xy-plane and the positive x-axis (θ), and the height of the point above the xy-plane (z).

## 2. How are cylindrical coordinates related to Cartesian coordinates?

Cylindrical coordinates can be converted to Cartesian coordinates using the following equations: x = rcos(θ), y = rsin(θ), and z = z. This means that a point in space can be described using both cylindrical and Cartesian coordinates, and they are simply two different ways of representing the same point.

## 3. What is a triple integral in cylindrical coordinates?

A triple integral in cylindrical coordinates is an extension of a double integral, where we integrate over a volume in three-dimensional space. It is represented by ∭f(r, θ, z) dV, where the limits of integration are determined by the shape of the volume and the function being integrated.

## 4. How do you set up a triple integral in cylindrical coordinates?

To set up a triple integral in cylindrical coordinates, we first determine the limits of integration for each variable (r, θ, and z) based on the shape of the volume. We then replace dV with r dr dθ dz and integrate the function f(r, θ, z) over the specified limits.

## 5. What are some common applications of cylindrical coordinates and triple integrals?

Cylindrical coordinates and triple integrals are commonly used in physics and engineering to solve problems involving three-dimensional objects, such as calculating the volume of a solid or finding the center of mass of an object. They are also used in calculus to evaluate difficult integrals that cannot be solved using Cartesian coordinates.

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