Kay L.

asked • 03/27/21# 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

*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 :)

## 1 Expert Answer

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∫_{0}^{15}(5+0.1e^{0.2x})^{2}dx

f^{2}(x) = 25 + e^{0.2x} + 0.01e^{0.4x}

V = π∫_{0}^{15} 25 + e^{0.2x} + 0.01e^{0.4x}dx = π[25x + 0.2e^{0.2x} + 0.004e^{0.4x}]_{0}^{15}

= π[(25(15) +0.2e^{3} + 0.01e^{6} -(0.2+0.004)]

≈ 488.5π cm^{3}

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Paul M.

03/27/21