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Date	May 2019	Marks available	3	Reference code	19M.2.SL.TZ2.S_2
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 2
Command term	Find	Question number	S_2	Adapted from	N/A

Question

Let $f\left( x \right) = 4 - 2{{\text{e}}^x}$ . The following diagram shows part of the graph of $f$ .

Find the $x$ -intercept of the graph of $f$ .

[2]

The region enclosed by the graph of $f$ , the $x$ -axis and the $y$ -axis is rotated 360º about the $x$ -axis. Find the volume of the solid formed.

[3]

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

valid approach (M1)

eg $f\left( x \right) = 0$ , $4 - 2{{\text{e}}^x} = 0$

0.693147

$x$ = ln 2 (exact), 0.693 A1 N2

[2 marks]

attempt to substitute either their correct limits or the function into formula (M1)

involving ${f^2}$

eg $\int_0^{0.693} {{f^2}}$ , $\pi {\int {\left( {4 - 2{{\text{e}}^x}} \right)} ^2}{\text{d}}x$ , $\int_0^{{\text{ln}}\,2} {{{\left( {4 - 2{{\text{e}}^x}} \right)}^2}}$

3.42545

volume = 3.43 A2 N3

[3 marks]

Examiners report

[N/A]

Syllabus sections

Topic 5—Calculus » SL 5.5—Integration introduction, areas between curve and x axis

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Topic 5—Calculus » AHL 5.12—Areas under a curve onto x or y axis. Volumes of revolution about x and y

Topic 5—Calculus

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