voume of region?

Question

The base of a solid S is the region enclosed by the graph of y=square root of (ln x) the line x=e and the x-axis. If the cross sections of s perpendicular to the x-axis are squares, then the volume of s is ?

Answer 1

Split the solid into elementary solids whose bases are slices of the original solid paralell to tyhe y-axis and whose height is dx. Then, the area of the base of each of such elemenray solids is the area of the square with side sqrt(ln(x)), so sqrt(ln(x)) ^2 = ln(x). It follows the volume of each elementary solid is dV = ln(x) dx.

To find the volume V our solid, we have to integrate the dv's from the point the curve intersects the x -axis to the point x =e. The intersection with the x axis is the point for which y = 0, that is x =1. So, we get V = Int (1 to e) ln(x) dx.

We know the primitive (one of them, considering constants of integration) of ln(x) is x* ln(x) - x. This can be easily done by parts, making u = ln(x) and dv = dx.

Therefore, V = [{x ln(x) - x] (1 to e) = [e*1 - e -(1*0 -1] = 1 unit of volume

Answer 2

Imagine the volume split into many slices parallel to the y axis. Each slice would be almost a very thin cuboid with length dx and width sqrt(lnx) and height sqrt(lnx). Work out the volume of such a slice as a function of x and then integrate it over the range 1 to e. The lower limit is 1 because the curve cuts the x axis at x = 1.