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

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?

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

If you put the wood on its side in the lathe, after cutting, this is like rotating the f(x) about the x-axis.

The Volume would be ∫_a^b π(f(x))² dx

For this case:

V = π∫₀¹⁵(5+0.1e^0.2x)²dx

f²(x) = 25 + e^0.2x + 0.01e^0.4x

V = π∫₀¹⁵ 25 + e^0.2x + 0.01e^0.4xdx = π[25x + 0.2e^0.2x + 0.004e^0.4x]₀¹⁵

= π[(25(15) +0.2e³ + 0.01e⁶ -(0.2+0.004)]

≈ 488.5π cm³