Integrals in cylindrical coordinates.

[cp255]

Integrate the function f(x,y,z)=−7x+2y over the solid given by the "slice" of an ice-cream cone in the first octant bounded by the planes x=0 and y=sqrt(263/137)x and contained in a sphere centered at the origin with radius 25 and a cone opening upwards from the origin with top radius 20.

I am not sure I am getting the right picture. Here are the bounds for the integral I found.

arctan(sqrt(263/137)) <= theta <= pi/2
0 <= z < 15
0 <= r <= (4/3)z

I am integrating in the order of dr dz d_theta.

The integrand I cam up with is...
-7r² * cos(theta) + 2r² * sin(theta) dr dz d_theta

Can anyone tell me where I went wrong. I keep getting a crazy answer.

 

[ehild]

You did it well, the bounds are correct. What is the result?

ehild

 

[cp255]

I finally did the integral right and I get 3500 * sqrt(263) + 1000 * sqrt(137) - 70000 which is about -1534.83. I put the exact answer into my web HW and it is wrong. I checked the integral with my CAS calculator and this is what it gets as well. Maybe the answer is positive but I only have six attempts and I don;t want to waste them. Can someone do the integral and tell me what they get?

 

[ehild]

Try to integrate from x=0 to the section of the line in the third quadrant, that is from theta=pi/2 to theta = pi+arctan(sqrt(263/137))

ehild

 

[cp255]

I think the problem said it is only in the volume in the first octant.

 

[ehild]

I see. Then your result must be correct, but use less significant digits. I would omit the decimals.

ehild

 

[cp255]

With this web HW system I can actually enter the exact value "3500 * sqrt(263) + 1000 * sqrt(137) - 70000".

 

[ehild]

Maybe it would do... ehild