Find the Volume (Double Integral)

[subflood]

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 y = x^{2} and the planes z = 3y, z = 2+y

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

 

[Big-T]

To find a volume I suppose you have to set up a triple integral?

You may, for example, let:
3y &lt; z &lt; 2 + y, \quad<br /> -\sqrt{y} &lt; x &lt; \sqrt{y} \quad,<br /> 0 &lt; y &lt; 1

.. I think, not sure, though.

 

[Pere Callahan]

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 y = x^{2} and the planes z = 3y, z = 2+y

I'm not even sure where to start. I have drawn the figure and understand that you have to integrate the two functions z = 3y and z = 2+y 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 0\leq y\leq \Gamma and -\sqrt{y}\leq x\leq\sqrt{y}, where \Gamma ist the y-value for which the two planes intersect.

So

<br /> \int_0^\Gamma{dy\int_{-\sqrt{y}}^{\sqrt{y}}dx(2y-2)}<br />