# Exam-Style Question on Integration

## A mathematics exam-style question with a worked solution that can be revealed gradually

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Question id: 699. This question is similar to one that appeared on an IB AA Higher paper in 2021. The use of a calculator is allowed.

A function $$f$$ is defined by $$f(x) = \frac{ke^{\frac{x}{2}}}{1 + e^x}$$ where $$x \in \mathbb{R}$$, $$x \geq 0$$ and $$k \in \mathbb{Z}^+$$.

The region enclosed by the graph of $$y = f(x)$$, the $$x$$-axis, the $$y$$-axis and the line $$x = 3$$ is rotated $$360^\circ$$ about the $$x$$-axis to form a solid of revolution.

(a) Show that the volume of the solid formed is $$\pi k^2 \left( \frac{1}{2} - \frac{1}{1 + e^3} \right)$$ cubic units.

(b) Find the minimum value of $$k$$ such that the volume is at least 90 cubic units.

Sue wants to make a small bowl with a volume of 90 cm3 based on the result from parts (a) and (b).

Sue investigates how the cross-sectional radius of the bowl changes.

(c) By considering the graph of a suitable derivative of $$f$$, find where the cross-sectional radius of the bowl is decreasing most rapidly.

(d) State the cross-sectional radius of the bowl at this point.

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