Cross section with equilateral triangles and integration

Question

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.

Answer 1

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.

Answer 2

If you look at the region of interest, you should see that the distance between the functions above and below the $x$-axis takes on the value of $y=e^x+2$ for $x\in[1,3]$:

This value is now the value of the base/side of the equilateral triangle that lies perpendicular to the $x$-axis; i.e. $S(x) = e^x+2$. Now, if you know the side $S(x)$, what is the area of an equilateral triangle in terms of $S(x)$? This will be the value of $A(x)$, the area of your cross section. The volume of the region would then be $V=\displaystyle\int_1^3A(x)\,dx$.

I hope you can take things from here.