The base of a solid is the region in the first quadrant bounded by the x-axis, the y-axis, the curve y = e^x, and the vertical line x = 3. Each cross section perpendicular to the x-axis is a square. Find the volume of the solid.

Step-by-step solution

1. 1

   Read the base one vertical slice at a time. Volume-by-cross-sections problems never need the shape of the whole solid, only the size of one slice. Fix an $x$ between $0$ and $3$ and slice the base vertically. The bottom of the base is the $x$-axis ($y=0$) and the top is the curve $y=e^{x}$, so that slice has length

   $L(x)=e^{x}-0=e^{x}.$

   The $y$-axis and the line $x=3$ only tell us where the slicing starts and stops: $0\le x\le 3$.

2. 2

   Turn the slice length into a cross-sectional area. The cross section standing on that slice is a square, and the slice is one full side of it. So the side length is $s=e^{x}$ and the area of the cross section is

   $A(x)=s^{2}=\left(e^{x}\right)^{2}=e^{2x}.$

   The rule $(e^{x})^{2}=e^{2x}$ is what makes this integral elementary, so the squaring should be done before integrating, not after.

3. 3

   Set up the volume integral. Stacking the slabs of area $A(x)$ and thickness $dx$ from $x=0$ to $x=3$ gives

   $V=\int_{0}^{3}A(x)\,dx=\int_{0}^{3}e^{2x}\,dx.$

4. 4

   Integrate. With the substitution $u=2x$ (or just by recognising the pattern), $\int e^{2x}\,dx=\tfrac12 e^{2x}$, because differentiating $\tfrac12 e^{2x}$ puts the factor $2$ back:

   $V=\left[\frac{1}{2}e^{2x}\right]_{0}^{3}=\frac{1}{2}e^{6}-\frac{1}{2}e^{0}.$

5. 5

   Evaluate and sanity-check. Since $e^{0}=1$,

   $V=\frac{e^{6}-1}{2}\approx 201.214.$

   A quick check: the largest cross section (at $x=3$) has area $e^{6}\approx 403.4$ and the smallest (at $x=0$) has area $1$, so a volume of about $201$ over a length of $3$ is the right order of magnitude — the areas grow so fast that almost all of the volume sits near $x=3$. Numerical integration of $e^{2x}$ on $[0,3]$ returns $201.21440$, matching the exact value.