Volume of Solids of Revolution With Known Cross Section

[aka757]

"The solid has a circular base with radius 1, and the cross sections perpendicular to a fixed diameter of the base are squares. Find the volume."

I know that the equation for the circle is x^2 + y^2 = 1 so since can be perpendicular to either axis, solving for "y" will yield y = +/- sqrt(1-x^2). I also know that eventually I will have to integrate the area of the cross section but I don't know what to do after solving for "y".

 

[tkhunny]

Methinks you are not understanding the question.

1) It is NOT a "Solid of Revolution". It's just a solid. The cross sections of solids of revolution tend to be circles or torii.

2) It matter not where you build your squares. Just pick an axis. The length of the chord controls the size of the square. At the Origin, the square is 2x2. Do you see why? The circle defines the bases of the squares. x = 0 ==> Base = 2, x = 1 ==> Base = 0, x = 1/2 ==> Base = sqrt(3)/2

3) There are symmetries. Exploit them. In other words, Don't do this: \(\displaystyle \int_{-1}^{1}f(x)\;dx\) Do this instead: \(\displaystyle 2 \cdot \int_{0}^{1}f(x)\;dx\)

Now what?

 

[aka757]

Apparently this is what my classmates (and my teacher) did:

Solve for y: y = +/- sqrt(1-x^2)
sqrt(1-x^2) - (-sqrt(1-x^2)) = 2sqrt(1-x^2)
Plug that into the equation for the cross section (square): (2sqrt(1-x^2))^2 = 4(1-x^2) = 4-4x^2
Integrate that from the interval [-1, 1] which eventually yields 16/3

The only step I don't understand is the 2nd one listed.

 

[tkhunny]

Step #2 is calculating the length of the base of each square. That shown is a generalized method that is a little much for a circle. It's a circle. Diameters bisect circles. The distance from the x-axis is \(\displaystyle \sqrt{1-x^{2}}\). That is 1/2 the base. You must see why in order to continue.