Using her lathe, a woodworker creates a wooden vase that is 15cm tall, with a side profile that can be described with the equation y=5+0.1e^0.2x. What is the volume of the vase?

Question

Using her lathe, a woodworker creates a wooden vase that is 15cm tall, with a side profile that can be described with the equation y=5 + 0.1e^0.2x

What is the volume of the vase?

Graph image:

https://cdn.discordapp.com/attachments/492994617156042752/825163857068687360/unknown.png

Note from teacher:

"The volume question here is total displacement including the thickness of the vase itself, not just the volume it can hold"

Any help is appreciated :)

Answer 1

This is a problem involving rotating the curve 360 degrees around the x axis. If you divide the function into slices from 0 to y and rotate that about the x axis, the volume of the slice would be

πy²Δx. Summing these from 0 to 15cm as Δx → 0 is the integral of πy²dx between 0 and 15 where y is the function given.

My integration skills have weakened, so maybe you can finish the problem.