Computing volume by cross section method.

Question

The base of the solid below is the region in the $xy$-plane bounded by the $x$-axis,the graph of y = $\sqrt{x}$ and the line $ x = 3 $. Find the volume of the solid.

Each cross-section of S perpendicular to the x-axis is a square with one side in the$ xy$-plane.

Answer 1

So here's the graph of the relevant portion of $f(x) = \sqrt x$:

And here is a pic of the resulting solid with the same portion shaded for comparison:

Imagine taking slices of this solid as you move along the x-axis from 0 to 3. At each point x, the cross-sectional area $A(x)$ is $(\sqrt x)^2$. Now integrate $A(x)$ over the interval $[0, 3]$, so $V = \int _0^3(\sqrt x)^2dx = \int_0^3 xdx = \frac 9 2$

Answer 2

Since you are given cross-sections perpendicular to the x-axis, your limits of integration and integrand will be in terms of x:

$V=\int_0^3 A(x) \;dx=\int_0^3(s(x))^2\;dx$ where, as noted in the first comment above, $s(x)=\sqrt{x}$.