Find the Volume (Double Integral)

Gamma^1{dy\int_{-\sqrt{y}}^{\sqrt{y}}dx(3y-2)} would be the integral you need.In summary, the problem involves finding the volume of a solid enclosed by a parabolic cylinder and two planes. The boundaries can be set up by finding the intersection point of the two planes and using it as the upper limit for the y-values. Then, the integral can be set up as a double integral with the appropriate functions and limits.
  • #1
subflood
2
0
I'm having trouble trying to setup this double integral. The question asks to find the volume of a solid enclosed by the parabolic cylinder [tex]y = x^{2}[/tex] and the planes [tex]z = 3y[/tex], [tex]z = 2+y[/tex]

I'm not even sure where to start. I have drawn the figure and understand that you have to integrate the two functions [tex]z = 3y[/tex] and [tex]z = 2+y[/tex] and subtract the volumes. However I'm stuck trying to setup the boundaries. Thanks.
 
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  • #2
To find a volume I suppose you have to set up a triple integral?

You may, for example, let:
[tex]3y < z < 2 + y, \quad
-\sqrt{y} < x < \sqrt{y} \quad,
0 < y < 1[/tex]

.. I think, not sure, though.
 
  • #3
subflood said:
I'm having trouble trying to setup this double integral. The question asks to find the volume of a solid enclosed by the parabolic cylinder [tex]y = x^{2}[/tex] and the planes [tex]z = 3y[/tex], [tex]z = 2+y[/tex]

I'm not even sure where to start. I have drawn the figure and understand that you have to integrate the two functions [tex]z = 3y[/tex] and [tex]z = 2+y[/tex] and subtract the volumes. However I'm stuck trying to setup the boundaries. Thanks.



If the problem is correct as you stated it the boundaries would be [itex]0\leq y\leq \Gamma[/itex] and [itex]-\sqrt{y}\leq x\leq\sqrt{y}[/itex], where [itex]\Gamma[/itex] ist the y-value for which the two planes intersect.

So

[tex]
\int_0^\Gamma{dy\int_{-\sqrt{y}}^{\sqrt{y}}dx(2y-2)}
[/tex]
 
Last edited:

What is the concept of volume in double integrals?

The concept of volume in double integrals is the measurement of the amount of space occupied by a three-dimensional object. In mathematics, it is calculated by taking the double integral of a function over a region in the xy-plane.

How do you find the volume using double integrals?

To find the volume using double integrals, you first need to determine the limits of integration for both the x and y variables. Then, you need to set up the integral using the appropriate function and limits. Finally, you can evaluate the integral using techniques such as substitution or integration by parts.

What are the applications of finding volume using double integrals?

Finding volume using double integrals has various applications in physics, engineering, and economics. It can be used to calculate the mass of an object with varying density, the center of mass of a solid, or the volume of a region in space. It is also useful in calculating probabilities and expected values in statistics and economics.

What are the differences between single and double integrals in finding volume?

The main difference between single and double integrals in finding volume is the number of variables involved. Single integrals involve only one variable, while double integrals involve two variables. Double integrals also require setting up the integral over a region in the xy-plane, whereas single integrals are usually set up over an interval on the x-axis.

What are some common mistakes when finding volume using double integrals?

Some common mistakes when finding volume using double integrals include using the wrong limits of integration, not accounting for the correct variables, and not choosing the appropriate function to integrate. It is also important to be careful with units and to check your answer for reasonableness. Additionally, forgetting to account for any holes or voids in the object can also lead to incorrect results.

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