Find volume of object if cross-sections are known

[ProBasket]

The base of a certain solid is an equilateral triangle with altitude 14. Cross-sections perpendicular to the altitude are semicircles. Find the volume of the solid, using the formula V= \int_a^{b} A(x)dx

applied to the picture shown attached to this post (click for a better view), with the left vertex of the triangle at the origin and the given altitude along the -axis.


i figure a= 0 and b = 14 and i will be plugging it into the formula pi*r^2 right?

the problem I'm having is finding the cross-section. can someone lend me a hand?

 

[Galileo]

You'll be integrating from x=0 to x=14 right?

First, find the length of one side of the equilateral, let's call it 2R.

It's clear at x=0, the radius is zero and at x=14 the radius is R, and the radius varies linearly with x. So set up an equation for the radius in terms of x.
then you can find A(x) and integrate.

 

[Jayboy]

Think of a volume of revolution formed by a general line y = mx. What sort of shape does this yield and how is it related to the shape you are given?

Thinking about this will also provide a good check as to whether your answer is correct as there is a nice formula for its volume.