Hello guys so I needed help with a problem which is:
Let $S$ be the solid with flat base, whose base is the region in the $xy$-plane defined by the curves $y=e^x$, $y=−2$, $x=1$ and $x=3$, and whose cross-sections perpendicular to the $x$-axis are equilateral triangles with bases that sit in the $xy$- plane.
Find the area $A(x)$ of the cross-section of $S$ given by the equilateral triangle that stands perpendicular to the $x$-axis, at coordinate $x$
The main problem I am having with this question is setting up the integral, I have not dealt with equilateral triangle and cross sections before so I am having a little trouble. Any help would be appreciated. Thanks in advance.
Best Answer
The problem merely seems to ask for the cross-sectional area $A(x)$. Note that, for an equilateral triangle of side $s$, the area is $\sqrt{3} s^2/4$. In this case, $s = e^x+2$, so that
$$A(x) = \frac{\sqrt{3}}{4} (e^x+2)^2$$
The volume calculation is simply an integral of $A(x)$ over $x \in [1,3]$, or
$$V = \frac{\sqrt{3}}{4} \int_1^3 dx\, (e^x+2)^2$$
which I imagine you can handle.