Finding the volume under a plane and region (polar coordinates)

In summary, the conversation is about trying to compute the volume of a region under a plane and over a circle in the xy-plane. The person is having trouble with the coordinates and equations, but with the help of HallsofIvy, they are able to solve the problem by changing coordinates and integrating.
  • #1
marc.morcos
11
0
Hey I am trying to compute the volume of the region under the plane z=7 x + 4 y + 34 and over the region in the xy -plane bounded by the circle x^2+y^2=4 y.

i can't seem to get it... like i i know that the circle is x^2+(x-2)^2=4
so 0<r<2 and 0<theta<2pi

this is what i try
double integral of (7x+4y+34) in polar tho... and it doesn't work... wat am i doinig wrong.. and i can't seem to center the circle... any help would be much appreciated...
 
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  • #2
marc.morcos said:
Hey I am trying to compute the volume of the region under the plane z=7 x + 4 y + 34 and over the region in the xy -plane bounded by the circle x^2+y^2=4 y.

i can't seem to get it... like i i know that the circle is x^2+(x-2)^2=4
so 0<r<2 and 0<theta<2pi
You mean, of course, x2+ (y-2)2= 4. More importantly, that region in the plane is NOT given by 0< r< 2, 0< [itex]\theta[/itex]< [itex]2\pi[/itex]. That describes a circle of radius 2 centered at (0,0)

this is what i try
double integral of (7x+4y+34) in polar tho... and it doesn't work... wat am i doinig wrong.. and i can't seem to center the circle... any help would be much appreciated...
Change coordinates. Let x'= x, y'= y- 2 so that (0, 2), the center of the circle in xy coordinates, becomes (0,0) in x'y' coordinates. Since x= x', y= y'+2, The equation of the circle is now x'2+ y'2= 4. Replace x, y in the equation of the plane by x'= x, y'= y+ 2 and integrate.
 
  • #3
thx a lot HallsofIvy, much appreciated!
 

Related to Finding the volume under a plane and region (polar coordinates)

1. What is the formula for finding the volume under a plane and region in polar coordinates?

The formula for finding the volume under a plane and region in polar coordinates is V = ∫∫r dr dθ, where r represents the radius and θ represents the angle in polar coordinates.

2. How do you determine the region in polar coordinates for finding the volume?

To determine the region in polar coordinates for finding the volume, you need to set up the limits of integration for both r and θ. This can be done by graphing the polar equation and identifying the boundaries of the region on the graph.

3. Can the volume under a plane and region in polar coordinates be negative?

No, the volume under a plane and region in polar coordinates cannot be negative. This is because the limits of integration for both r and θ are always positive values, resulting in a positive volume.

4. What is the difference between finding the volume under a plane and region in polar coordinates and Cartesian coordinates?

The main difference between finding the volume under a plane and region in polar coordinates and Cartesian coordinates is the method of integration. In polar coordinates, the volume is found by integrating with respect to r and θ, while in Cartesian coordinates, the volume is found by integrating with respect to x, y, and z.

5. Can the volume under a plane and region in polar coordinates be calculated for irregular shapes?

Yes, the volume under a plane and region in polar coordinates can be calculated for irregular shapes as long as the boundaries of the region can be determined and the polar equation can be integrated to find the volume. However, it may be more difficult to set up the limits of integration in this case.

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