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Date May 2012 Marks available 9 Reference code 12M.2.hl.TZ1.8
Level HL only Paper 2 Time zone TZ1
Command term Deduce Question number 8 Adapted from N/A

Question

A cone has height h and base radius r . Deduce the formula for the volume of this cone by rotating the triangular region, enclosed by the line y=hhrxy=hhrx and the coordinate axes, through 2π2π about the y-axis.

Markscheme

x=rrhy or x=rh(hy) (or equivalent)x=rrhy or x=rh(hy) (or equivalent)     (A1)

πx2dyπx2dy

=πh0(rrhy)2dy=πh0(rrhy)2dy     M1A1 

Note: Award M1 for x2dyx2dy and A1 for correct expression.

Accept πh0(rhyr)2dy and πh0(±(rrhx))2dxπh0(rhyr)2dy and πh0(±(rrhx))2dx

 

=πh0(r22r2hy+r2h2y2)dy=πh0(r22r2hy+r2h2y2)dy     A1

Note: Accept substitution method and apply markscheme to corresponding steps.

 

=π[r2yr2y2h+r2y33h2]h0=π[r2yr2y2h+r2y33h2]h0     M1A1 

Note: Award M1 for attempted integration of any quadratic trinomial.

 

=π(r2hr2h+13r2h)=π(r2hr2h+13r2h)     M1A1 

Note: Award M1 for attempted substitution of limits in a trinomial.

 

=13πr2h=13πr2h     A1 

Note: Throughout the question do not penalize missing dx/dy as long as the integrations are done with respect to correct variable.

 

[9 marks]

Examiners report

Most candidates attempted this question using either the formula given in the information booklet or the disk method. However, many were not successful, either because they started off with the incorrect expression or incorrect integration limits or even attempted to integrate the correct expression with respect to the incorrect variable.

Syllabus sections

Topic 6 - Core: Calculus » 6.5 » Volumes of revolution about the xx-axis or yy-axis.
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