Triple integral in cylindrical coordinates

[Calpalned]

Homework Statement

Evaluate $\int \int \int_E {x}dV$ where E is enclosed by the planes $z=0$ and $z=x+y+5$ and by the cylinders $x^2+y^2=4$ and $x^2+y^2=9$.

Homework Equations

$\int \int \int_E {f(cos(\theta),sin(\theta),z)}dzdrd \theta$
How do I type limits in for integration?

The Attempt at a Solution

Right now I'm just trying to find the limits of integration.

For dz, $z=x+y+5$ is equivalent to $z=rcos \theta +r sin \theta +5$ so that is the upper limit for $z$ while $z=0$ is the lower limit.

For dθ I am going to assume that it is $0 < \theta < 2 \pi$ By the way, how do I make the "greater than or equal to" sign in Latex? I choose zero and two pi because question didn't say the object is restricted to any octant. Keep in mind that I do not know what the graph looks like visually so I am taking a risk...

For dr, I have $r^2 = 4$ and $r^2 = 9$ what do I do? Additionally, can the lower limit for r ever be negative?

 

[Mark44]

Homework Statement

Evaluate $\int \int \int_E {x}dV$ where E is enclosed by the planes $z=0$ and $z=x+y+5$ and by the cylinders $x^2+y^2=4$ and $x^2+y^2=9$.

Homework Equations

$\int \int \int_E {f(cos(\theta),sin(\theta),z)}dzdrd \theta$

In polar and cylindrical coordinates you're going to need an extra factor of r, as in $r~dr~d\theta$, which is equivalent to dx dy in rectangular coordinates.

How do I type limits in for integration?

# # \int_{a}^{b} ... # #
Here a is the lower limit and b is the upper limit. If either limit is a single character, you don't need the braces. However, if a limit consists of two or more characters, like -3, then you need the braces.

The Attempt at a Solution

Right now I'm just trying to find the limits of integration.

For dz, $z=x+y+5$ is equivalent to $z=rcos \theta +r sin \theta +5$ so that is the upper limit for $z$ while $z=0$ is the lower limit.

For dθ I am going to assume that it is $0 < \theta < 2 \pi$ By the way, how do I make the "greater than or equal to" sign in Latex?

\ge for ≥ and \le for ≤

I choose zero and two pi because question didn't say the object is restricted to any octant. Keep in mind that I do not know what the graph looks like visually so I am taking a risk...

Then you should sketch a graph of your region. It's not that complicated. The region is a hollow tube whose walls are formed by the two cylinders, with the cap being the plane z = x + y + 5. This plane makes a diagonal slice through the tube.

For dr, I have $r^2 = 4$ and $r^2 = 9$ what do I do? Additionally, can the lower limit for r ever be negative?

You can take r = 2 and r = 3. It's possible for r to be negative, due to the ambiguity of coordinates in polar and cylindrical coordinates. By that I mean that a single point can have multiple representations, which is not the case in rectangular coordinates. For example (-1, 7π/6) is the same point as (1, π/6).