How to Evaluate the Volume Bounded by Given Surfaces?

In summary: I am not sure where the "right side" is. I am thinking it must be on the inside of the cylinder but I am not sure where.
  • #1
Alex_Neof
41
2

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?
 
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  • #2
Alex_Neof said:

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.
 
  • #3
SteamKing said:
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.
 
  • #4
Have you at least sketched the region? It is bounded below by the xy-plane, z= 0, above by [itex]z= 2+ x^2[/itex], and on the side by the cylinder [itex]x^2+ y^2= a^2[/itex]. From any point in the xy-plane, inside that cylinder, the height is [itex]2+ x^2- 0= 2+ x^2[/itex]. Convert that to cylindrical coordinates.
 
  • #5
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.
 
  • #6
Oh thank you guys for your replies. Zondrina, the problem I am having is determining the limits for Z.
 
  • #7
[itex]x^2+ y^2= a^2[/itex] does not involve z so is the cylinder forming the sides. The "top" and "bottom" must be given by [itex]z= 2+ x^2[/itex] and the "xy- plane" which is z= 0,
 

Related to How to Evaluate the Volume Bounded by Given Surfaces?

What is bounded volume?

Bounded volume refers to the amount of space that an object occupies within a defined boundary or enclosure.

Why is evaluating bounded volume important?

Evaluating bounded volume is important because it allows scientists to accurately measure and compare the size or capacity of objects. This information can be used to make predictions, conduct experiments, or determine the suitability of an object for a particular purpose.

How is bounded volume measured?

Bounded volume is typically measured using mathematical formulas that calculate the volume of a three-dimensional shape, such as a cube or cylinder. Other methods include using water displacement or specialized tools such as calipers or rulers.

What are some common units of measurement for bounded volume?

The most commonly used units of measurement for bounded volume are cubic units, such as cubic meters, cubic feet, or cubic centimeters. Other units, such as liters or gallons, are also used depending on the object being measured.

How can the accuracy of bounded volume measurements be improved?

The accuracy of bounded volume measurements can be improved by using precise and calibrated measuring tools, taking multiple measurements and averaging them, and ensuring that the boundary or enclosure is well-defined and consistent.

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