Volumes: Using 3d objects as slice question? (1 Viewer)

[kayven]

Hey guys,

Can you please solve this question for me?
It goes on the lines of: The base of a circle has the equation x^2 + y^2 = 9. Find the volume of the solid if every cross section perpendicular to the y-axis is a square with one side in the base of the circle.


Thankyou so much!

 

[Carrotsticks]

You will need to draw 2 diagrams.

1. The circle with radius 3, centred at the origin. Shade a thin vertical slice (or horizontal, but I like vertical because horizontal looks fat) which has thickness delta(x) and its length/height would be 2y.

2. A square which would have side length 2y, as above.

The area of the slice is (2y)(2y) = 4y^2. However, it has thickness delta(x) so the volume of our slice is 4y^2 * delta(x).

Now, we integrate the expression for our slice from x=-3 to x=3 so our integral becomes:



But we have to integrate with respect to X because we have dx. Our expression is in terms of y, so we have to change it back in terms of x by perhaps using the equation of the circle x^2+y^2=9.

So our integral now is:



Then evaluate it as usual.

 

[kayven]

Thanks Carrotsticks, but shouldnt the thickness be delta y considering that the slice is taken perpendicular to the y=axis?

 

[Carrotsticks]

Thanks Carrotsticks, but shouldnt the thickness be delta y considering that the slice is taken perpendicular to the y=axis?

Not quite. But actually, the question can be done either way. Your slice would even be in the form y=mx+b because of the rotational symmetry of the circle! But of course that would make things unnecessarily complicated.