8.8 Volumes with Cross Sections: Triangles and Semicircles

This topic is very similar to topic 8.7 except now we’re using triangles and semicircles as cross sections. To find the area of a shape using a triangle or semicircle cross section, take a look at the example below. 

Let’s say we are asked to find the volume of a solid whose base is the circle x^2 + y^2 = 4, where the cross sections perpendicular to the x-axis are all equilateral triangles. NOTE: Sometimes you'll be given isosceles right triangles or semicircles. Make sure you using the right formula!

We find the side of the triangle by doing top minus bottom. Thus, the side of the triangle is 2√4 - x^2. Now, we use the formula for the equilateral triangle to find the area of the cross section (A = (side)2 *  √3 / 4). We can now find the volume by integrating the area function from x = -2 to x = 2.