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

[marc.morcos]

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...

 

[HallsofIvy]

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< \theta< 2\pi. 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.

 

[marc.morcos]

thx a lot HallsofIvy, much appreciated!