Need help finding the volume of a Angle Food Cake pan using integration.

[alexf322]

For class I had to measure an angle food cake pan using integration. The pan itself had a circular hole in the middle of it. So I figured I would take find the volume of the Large pan and then subtract the volume of the Small circular hole in the middle.

As far as integration goes, I am kind of lost. I measured the pan and found the height to be 10cm, bottom radius to be 10.5cm and the top radius to be 14.25cm. The hole in the middle of the pan has a height of 10cm, bottom radius of 3cm and a top radius of 2cm. I figured I would use the integration of a frustum of a cone around the y-axis but I am having a big trouble setting it up.

Any help would be appreciated.

 

[Mark44]

I think you mean angel food cake. Angle food cakes can get stuck in the throat if the angles are too acute.

 

[alexf322]

hah yeah I meant Angel Food Cake*

 

[HallsofIvy]

For class I had to measure an angle food cake pan using integration. The pan itself had a circular hole in the middle of it. So I figured I would take find the volume of the Large pan and then subtract the volume of the Small circular hole in the middle.

As far as integration goes, I am kind of lost. I measured the pan and found the height to be 10cm, bottom radius to be 10.5cm and the top radius to be 14.25cm. The hole in the middle of the pan has a height of 10cm, bottom radius of 3cm and a top radius of 2cm. I figured I would use the integration of a frustum of a cone around the y-axis but I am having a big trouble setting it up.

Any help would be appreciated.

If you are allowed to use the "frustrum of a cone" formula, you don't need to integrate at all. Find the volume of the frustrum of a cone ignoring the hole, find the volume of the hole, and subtract.

If you cannot use the formula but must integrate, note that if we set up a coordinate system having the origin in the center of the pan, the positive y-axis up and the positive x-axis to the right with units in cm. In that coordinate system, one side of the pan, on the right, passes through the points (10.5, 0) and (14.25, 10). Find the equation of that line, y= ax+ b. Then the radius of a disk, at height y, is x= (y- b)/a. Find the area of such a disk and integrate that area, with respect to y, from y= 0 to y= 10. Do the same to find the volume of the hole, and subtract.

 

[alexf322]

Also if I flipped the frustum over, so that the height of the frustum is laying on the x-axis, I can just integrate it over the x-axis right?