[Calculus] Volume of Solids by Cross Sections
There is a solid whose base is bounded by y=4-x^2 and the x-axis. Cross sections perpendicular to the base and parallel to the x-axis are equilateral triangles. Find the volume.
The answer is 8√3, but for some reason I haven't been getting that.
Posted Jan 13, 2019, 5:28 PM UTC
24
Step 1: Recognize that if the triangles are parallel to the x-axis, we'll be integrating with respect to y (dy), so the equation needs to be solved for y. You should get y=±√(4-y). To simplify, we can say that the side length of each triangle will be 2√(4-y).
Step 2: Look up (or derive) the area equation for equilateral triangles. It's A=√(3)s^(2)/4, where s is the side length.
Step 3: Integrate that area formula from y=0 to y=4, substituting the side length, s=2√(4-y).
V=√(3)∫(4-y)dy
V=4√(3)∫dy-√(3)∫(y)dy
V=4√(3)(4-0)-√(3)(4^(2)/2-0^(2)/2)
V=16√(3)-8√(3)
V=8√(3)
Thank you for the help! You got me unstuck on the homework.