1. ## Volume

The base of a solid is the region bounded by the parabola $\displaystyle y=\frac {1}{2}x^2$ and the line y = 2. Each plane section of the solid perpendicular to the y-axis is an equilateral triangle. Find the volume of the solid.

I did not understand the line "Each plane ............ equilateral triangle"
What shape does the solid have?
I have drawn the graph, but volume of which solid is it asking?

2. Originally Posted by Shyam
The base of a solid is the region bounded by the parabola $\displaystyle y=\frac {1}{2}x^2$ and the line y = 2. Each plane section of the solid perpendicular to the y-axis is an equilateral triangle. Find the volume of the solid.

I did not understand the line "Each plane ............ equilateral triangle"
What shape does the solid have?
I have drawn the graph, but volume of which solid is it asking?
a volume whose cross sections perpendicular to the y-axis are equilateral triangles ... it doesn't have a special name.

$\displaystyle V = \int_c^d A(y) \, dy$

$\displaystyle A = \frac{\sqrt{3}}{4} s^2$ , where $\displaystyle s = 2x = 2\sqrt{2y}$

$\displaystyle V = \sqrt{3} \int_0^2 2y \, dy$

3. different base, but check out the link for a good visualization ...

http://mathdemos.gcsu.edu/mathdemos/...ross75slab.gif

4. Originally Posted by skeeter
different base, but check out the link for a good visualization ...

http://mathdemos.gcsu.edu/mathdemos/...ross75slab.gif
$\displaystyle Area \;A = \frac{1}{2}(2x)(\sqrt 3 x)=x^2 \sqrt3$
$\displaystyle dV = A dy$
$\displaystyle V = \int\limits_0^2 {x^2 \sqrt 3 {\text{ }}dy} = \int\limits_0^2 {2y\sqrt 3 {\text{ }}dy} = 4\sqrt 3$