Cross section problems

[J6JA]

The base of a solid is bounded by y=2-.5x and the x & y axes. What is the volume if every cross section perpendicular to the x axis is a semicircle?


I thought the answer was (8/3)(pi), is it?
also
A solid has a circular base of radius 3. If every cross section perpendicular to the x axis is an equilateral triangle, then the volume =?

I wrote the EQ x^2 + y^2 = 9 and solved for y. (y= the root of 9-x^2) I know the Area of an equlateral triangle = (s^2)(root 3 divided by 4) Then I plugged in the y= the root of 9-x^2 in for s in the area EQ and then took the integral of it from -3 to 3, but I got 4.5 * root 3 which isn't right? What did I do wrong?

Thanks everyone for your help
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[Gene]

1) Since all you ask for was is it right, no. You don't show any work. Look at it with YZ plane as the base. You have 1/2 of a circular cone of radius y/2 = 1.
v=(1/3)bh =
(1/3)(pi*1^2/2)*(4) =
2pi/3

2) Did you forget that s = 2y, not y? Its a right circular cone of height 3sqrt(3), radius 3 so you are aiming for 9*pi*sqrt(3)

 

[galactus]

I apologize if I am not seeing it, but doesn't it have radius 2?.

\(\displaystyle \frac{1}{3}(\frac{{\pi}}{2})(2^{2})(4)=\frac{8{\pi}}{3}\)

\(\displaystyle \frac{{\pi}}{2}\int_{0}^{4}({2-\frac{x}{2}})^{2}dx=\frac{8{\pi}}{3}\)

\(\displaystyle {\pi}\int_{0}^{2}y(4-2y)dy=\frac{8{\pi}}{3}\)

 

[Gene]

As I see it, the semicircle diameter is from (0,0) to (0,2) with one side being the x axis. If it had a radius of 2 it would slop over to (0,-2) and would no longer be limited by the x axis.
-------------
Gene

 

[galactus]

OK Gene, but here's what I was picturing:

Of course, we are only dealing with the upper half, so the volume would be one-

half this.

 

[Gene]

Look at

The base of a solid is bounded by y=2-.5x and the x & y axes.

Then look at your latest picture and tell me how it is bounded by the x axis. It surounds the x axis.
Then go back to your first picture. The y axis between (0,0) and (0,2) is the top view of the semi circle with the center at (0,1,0). Three points on it are (0,0,0), (0,1,1) and (0,2,0). It is a radius 1 semi circle in the yz plane. That is bounded by the line and the x axis. Each of them is a line from the peak to the base ending at the ends of the diameter.

 

[galactus]

OK Gene. Sorry. I'll shut up about it now. I had a mental block. It's obvious now. :lol: