How to Evaluate the Volume Bounded by Given Surfaces?

[Alex_Neof]

Homework Statement

Using a suitable Jacobian, evaluate the volume bounded by the surface $z = 2 +x^2$, the cylinder $x^2 + y^2 = a^2$ (where $a$ is a constant), and the $x-y$ plane.

Homework Equations

$x = r cos{\theta}$
$y = r sin{\theta}$

The Attempt at a Solution

I determined the Jacobian to be $r$.
The limits for $\theta$ would be from $0$ to $2 \pi$.
The limits for $r$ would be from $0$ to $a$.

Could anyone kindly guide me through the problem?

 

[SteamKing]

Homework Statement

Using a suitable Jacobian, evaluate the volume bounded by the surface $z = 2 +x^2$, the cylinder $x^2 + y^2 = a^2$ (where $a$ is a constant), and the $x-y$ plane.

Homework Equations

$x = r cos{\theta}$
$y = r sin{\theta}$

The Attempt at a Solution

I determined the Jacobian to be $r$.
The limits for $\theta$ would be from $0$ to $2 \pi$.
The limits for $r$ would be from $0$ to $a$.

Could anyone kindly guide me through the problem?

Well, you went to the trouble to calculate a Jacobian for this problem, now what do you do with it?

There must be some formula where the Jacobian appears.

 

[Alex_Neof]

Well, you went to the trouble to calculate a Jacobian for this problem, now what do you do with it?

There must be some formula where the Jacobian appears.

Hi there SteamKing. I will use the Jacobian when I evaluate the integral.

$\int_{r_1}^{r_2} \int_{\theta_1}^{\theta_2} \int_{z_1}^{z_2} r \ dz d{\theta} dr$
something like that.

 

[HallsofIvy]

Have you at least sketched the region? It is bounded below by the xy-plane, z= 0, above by z= 2+ x^2, and on the side by the cylinder x^2+ y^2= a^2. From any point in the xy-plane, inside that cylinder, the height is 2+ x^2- 0= 2+ x^2. Convert that to cylindrical coordinates.

 

[STEMucator]

Using HallsofIvy's post, you need to use cylindrical co-ordinates to compute the integral and obtain the answer.

More formally, what you are doing is an integral transform by changing variables. You should have a change of variables theorem lying around telling you:

$$\iiint_D f(x, y, z) \space dV = \iiint_{D^\prime} f(r \text{cos}(\theta), r \text{sin}(\theta), z) \space J_{r, \theta}(x,y) \space dV' = \iiint_{D^\prime} f(r \text{cos}(\theta), r \text{sin}(\theta), z) \space J_{r, \theta}(x,y) \space dz d\theta dr$$

Where $J_{r, \theta}(x,y)$ is the Jacobian of the invertible transformation and $dV' = dz d\theta dr$ for cylindrical co-ordinates. For this particular problem:

$$J_{r, \theta}(x,y) = \left| \begin{array}{cc}
x_r & x_{\theta} \\
y_r & y_{\theta} \\
\end{array} \right| \quad \quad

x = r \text{cos}(\theta), y = r \text{sin}(\theta)$$

You have already found $J_{r, \theta}(x,y) = r$. You have also found the limits for $r$ and $\theta$ already. What about the limits for $z$? What does $z = 2 + x^2$ look like? It looks like it's bounded below by something.

 

[Alex_Neof]

Oh thank you guys for your replies. Zondrina, the problem I am having is determining the limits for Z.

 

[HallsofIvy]

x^2+ y^2= a^2 does not involve z so is the cylinder forming the sides. The "top" and "bottom" must be given by z= 2+ x^2 and the "xy- plane" which is z= 0,