How Do You Determine the Height in Volume Calculations by Integration?

[ada0713]

http://img408.imageshack.us/my.php?image=picture20017hu6.jpg

The attachement contains both problem and the attemp at a solution.
Specifically, I am stuck on finding the equation for the hight for each solid.
(height "a" for both solids)
Should I just set up a random letter to indicate the height
or is there any way that I can figure out the equation for it?

 

[rocomath]

Link it somewhere like imageshack, take like a day to get your attachment approved!

 

[ada0713]

Can you see it now?

 

[rohanprabhu]

take the 'O' as origin. Now, 'scan' through the x-axis.. like.. take a distance 'x' from the origin. From there, assume a cuboid of thickness 'dx' and find the area of that and then integrate within limits. The problem is that to find the width and length of the cuboid, you're going to need the equation of the curve shown there.. which doesn't seem to be given or since the scan is small.. i can't see it.

 

[ada0713]

I think the curve should be a circle becuase the radius is indicated as "r" in the picture.

 

[<---]

I think he means the plane curve, as you will need to know how the slope rises and falls to know the volumes of your pieces.

 

[rohanprabhu]

I think the curve should be a circle becuase the radius is indicated as "r" in the picture.

oh.. ur right mate.. should've figured that out :d

neways.. now that you have a circle, you can easily determine what the length of the cuboid will be at a distance of 'x' from the origin along the line. Also, the breadth is 'dx' and you can use ur image to get the height of the cuboid for a particular 'x'. [since the angle 'α' remains constant, you have a relation between 'x' and the height of the cuboid].

Find that and integrate within proper limits.. you should get the answer. Post here in case you have any problems.

 

[ada0713]

Okay.. I think I got the second one (with right triangle slice)
The "base" of the triangle should be sqrt((R^2-x^2)) (<- I got this from x^2+y^2=R^2)
and the "height" of the trangle is going to be sqrt((R^2-x^2))*tanα
(since tanα = height/base.. and you know "base" from above.. so you solve for height)

Than I pluged in two equations into
V=1/2*base*height*dx
and ended up with V= 1/2 (R^2-x^2)tanα dx

The rest of the integrating is not a big deal.. so i'll skip:)

Sorry that my work is kind of messy. I should have scanned my work
but I don't have a scanner with me at the moment..

 

[ada0713]

But the first one seems to be more complicated..
how do I find height of the rectangle?