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Exam-Style Question on Integration

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

List Of Questions Exam-Style Question More Integration Questions More on this Topic

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.

Small Bowl

(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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If you need more practice try the self-checking interactive exercises called Integration.

The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.

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