Calculus Disk/Washer Method for Volume

Question

I am given the bounded functions y=ln(x), g(x)=-.5x+3, and the x-axis. The reigon R is bounded between these, and I'm tasked with finding the volume of this solid using disk/washer method when revolved around the x-axis.

I know the formula I need to use, but I'm a little confused on finding the upper and lower limits and which to place as an innner and outer radius since the functions aren't graphed like the traditional washer method problem.

If someone could help with the definite integral for this problem that would be great!

Answer 1

If allowed, graphing the functions will tell you what you need to know to. In this case the graph looks like this (graphed in Desmos):

The region you described is the shaded region in the graph. Note the three intersection points. These points are of interest:

1. $x_0$, the intersection of $y = ln(x)$ and $y=0$ (x-axis)
2. $x_1$, the intersection of $y = ln(x)$ and $y=-0.5x+3$
3. $x_2$, the intersection of $y= -0.5x+3$ and $y=0$

This splits the region into two subregions.

1. The left subregion from $x=x_0$ to $x=x_1$ (i.e. $x_0 \leq x \leq x_1$). Also $0 \leq y \leq ln(x)$.
2. The right subregion from $x=x_1$ to $x=x_2$ (i.e. $x_1 \leq x \leq x_2$). Also $0 \leq y \leq -0.5x+3$.

Correspondingly, you need to perform two definite integrals (one for each subregion). Let $r$ denote the inner raidus and $R$ denote the outer radius.

1. The first definite integral has a limits of integration $x_0$ and $x_1$ and uses $r = 0$ and $R = ln(x)$
2. The second definite integral has a limits of integration $x_1$ and $x_2$ and uses $r = 0$ and $R = -0.5x+3$